[{"type":"mcq","problem_latex":"What is the vertex of the parabola defined by the equation $y = 3(x - 4)^2 + 1$?","problem_text":"What is the vertex of the parabola defined by the equation y = 3(x - 4)^2 + 1?","options_latex":["$(3, 1)$","$(4, -1)$","$(-4, 1)$","$(4, 1)$"],"options_text":["(3, 1)","(4, -1)","(-4, 1)","(4, 1)"],"answer_latex":"$(4, 1)$","answer_text":"(4, 1)","answerIndex":3,"explanation":"The equation $y = 3(x - 4)^2 + 1$ is in the standard vertex form for a parabola, $y = a(x - h)^2 + k$. In this form, the vertex is always located at the point $(h, k)$. By comparing the given equation to the vertex form, we see that $h = 4$ and $k = 1$. Thus, the vertex of the parabola is $(4, 1)$."},{"type":"mcq","problem_latex":"Determine the equation for the axis of symmetry for the quadratic function $y = -2(x + 5)^2 - 9$.","problem_text":"Determine the equation for the axis of symmetry for the quadratic function y = -2(x + 5)^2 - 9.","options_latex":["$x = 5$","$y = -5$","$x = -9$","$x = -5$"],"options_text":["x = 5","y = -5","x = -9","x = -5"],"answer_latex":"$x = -5$","answer_text":"x = -5","answerIndex":3,"explanation":"The quadratic function $y = -2(x + 5)^2 - 9$ is presented in vertex form, $y = a(x - h)^2 + k$. In vertex form, the axis of symmetry is the vertical line defined by $x = h$. To find $h$, we compare the given equation to the general form: $y = -2(x - (-5))^2 - 9$. Therefore, $h = -5$. The equation for the axis of symmetry is $x = -5$."},{"type":"mcq","problem_latex":"The graph of the function $y = -0.5(x - 2)^2 + 7$ opens in which direction?","problem_text":"The graph of the function y = -0.5(x - 2)^2 + 7 opens in which direction?","options_latex":["To the right","To the left","Upwards","Downwards"],"options_text":["To the right","To the left","Upwards","Downwards"],"answer_latex":"Downwards","answer_text":"Downwards","answerIndex":3,"explanation":"The given function $y = -0.5(x - 2)^2 + 7$ is in vertex form $y = a(x - h)^2 + k$. The direction the parabola opens is determined by the sign of the leading coefficient $a$. In this equation, $a = -0.5$. Since $a$ is negative ($a < 0$), the graph opens downwards."},{"type":"mcq","problem_latex":"For the equation $y = (x - 10)^2 - 3$, what are the values of $h$ and $k$ in the vertex form $y = a(x-h)^2 + k$?","problem_text":"For the equation y = (x - 10)^2 - 3, what are the values of h and k in the vertex form y = a(x-h)^2 + k?","options_latex":["$h=1, k=-3$","$h=10, k=-3$","$h=-10, k=-3$","$h=10, k=3$"],"options_text":["h=1, k=-3","h=10, k=-3","h=-10, k=-3","h=10, k=3"],"answer_latex":"$h=10, k=-3$","answer_text":"h=10, k=-3","answerIndex":1,"explanation":"The vertex form of a parabola is $y = a(x-h)^2 + k$. The given equation is $y = (x - 10)^2 - 3$. Comparing this to the vertex form, we identify the values. The term $(x - h)^2$ corresponds to $(x - 10)^2$, meaning $h=10$. The term $+k$ corresponds to $-3$, meaning $k=-3$."},{"type":"mcq","problem_latex":"Which of the following equations is written in the vertex form $y = a(x-h)^2 + k$?","problem_text":"Which of the following equations is written in the vertex form y = a(x-h)^2 + k?","options_latex":["$y = -5(x + 2)^2 + 8$","$y = 2(x + 3)(x - 1)$","$y = x^2 - 4x + 4$","$y = \\frac{1}{x-1} + 2$"],"options_text":["y = -5(x + 2)^2 + 8","y = 2(x + 3)(x - 1)","y = x^2 - 4x + 4","y = 1/(x-1) + 2"],"answer_latex":"$y = -5(x + 2)^2 + 8$","answer_text":"y = -5(x + 2)^2 + 8","answerIndex":0,"explanation":"The standard vertex form for a parabola is $y = a(x-h)^2 + k$. The equation $y = -5(x + 2)^2 + 8$ adheres to this structure. Comparing the forms, we identify $a = -5$, $h = -2$ (since $x+2 = x-(-2)$), and $k = 8$."},{"type":"mcq","problem_latex":"Identify the vertex $(h, k)$ for the parabolic function defined by $f(x) = -3 - 2(x + \\frac{1}{2})^2$.","problem_text":"Identify the vertex $(h, k)$ for the parabolic function defined by $f(x) = -3 - 2(x + \\frac{1}{2})^2$.","options_latex":["(2, 3)","$(\\frac{1}{2}, -3)$","$(-\\frac{1}{2}, -3)$","$(-\\frac{1}{2}, 3)$"],"options_text":["(2, 3)","$(\\frac{1}{2}, -3)$","$(-\\frac{1}{2}, -3)$","$(-\\frac{1}{2}, 3)$"],"answer_latex":"(-\\frac{1}{2}, -3)","answer_text":"(-1/2, -3)","answerIndex":2,"imageUrl":"","imageZoom":"100%","explanation":"The function is given in the standard vertex form of a parabola, $f(x) = a(x - h)^2 + k$, where the vertex is $(h, k)$. First, rearrange the given function $f(x) = -3 - 2(x + \\\\frac{1}{2})^2$ into the standard structure: $f(x) = -2(x + \\\\frac{1}{2})^2 - 3$. To find $h$, we compare $x - h$ with $x + \\\\frac{1}{2}$. Since $x - h = x + \\\\frac{1}{2}$, we take the opposite sign of the value inside the parentheses, so $h = -\\\\frac{1}{2}$. The $k$ value is the constant term added or subtracted outside the squared term, so $k = -3$. Therefore, the vertex $(h, k)$ is $(-\\\\frac{1}{2}, -3)$."},{"type":"mcq","problem_latex":"A parabola is given by the equation $g(x) = 5(2x - 6)^2 + 10$. Determine the coordinates of the vertex $(h, k)$.","problem_text":"A parabola is given by the equation g(x) = 5(2x - 6)^2 + 10. Determine the coordinates of the vertex (h, k).","options_latex":["(3, 10)","(\\frac{1}{2}, 10)","(6, 10)","(-6, 10)"],"options_text":["(3, 10)","(1/2, 10)","(6, 10)","(-6, 10)"],"answer_latex":"(3, 10)","answer_text":"(3, 10)","answerIndex":0,"explanation":"The standard vertex form of a parabola is $g(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex. To use this form, the term inside the parenthesis must isolate $(x - h)$. Start with the given equation: $g(x) = 5(2x - 6)^2 + 10$. Factor out the coefficient of $x$ from the inner parenthesis: $2x - 6 = 2(x - 3)$. Substitute this back into the function: $g(x) = 5[2(x - 3)]^2 + 10$. Apply the exponent to both factors inside the brackets: $g(x) = 5 \\\\cdot 2^2 (x - 3)^2 + 10$. Simplify the coefficients: $g(x) = 5 \\\\cdot 4 (x - 3)^2 + 10$, which results in the standard form $g(x) = 20(x - 3)^2 + 10$. By comparison with $g(x) = a(x - h)^2 + k$, we identify $h = 3$ and $k = 10$. The vertex is $(3, 10)$."},{"type":"mcq","problem_latex":"Find the vertex $(h, k)$ of the function $y = 4 [ (x+5)^2 - 1 ] + 7$.","problem_text":"Find the vertex (h, k) of the function y = 4 [ (x+5)^2 - 1 ] + 7.","options_latex":["(-5, -3)","(5, 3)","(-1, 7)","(-5, 3)"],"options_text":["(-5, -3)","(5, 3)","(-1, 7)","(-5, 3)"],"answer_latex":"(-5, 3)","answer_text":"(-5, 3)","answerIndex":3,"explanation":"The standard vertex form for a parabola is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex. Simplify the given function $y = 4 [ (x+5)^2 - 1 ] + 7$ by distributing the 4: $y = 4(x+5)^2 - 4(1) + 7$. Combine the constant terms: $y = 4(x+5)^2 + 3$. Comparing $y = 4(x+5)^2 + 3$ to $y = a(x - h)^2 + k$, we identify the vertex components. Since $x - h = x + 5$, we have $h = -5$. The constant term is $k = 3$. Therefore, the vertex $(h, k)$ is $(-5, 3)$."},{"type":"mcq","problem_latex":"The equation of a parabola is given in a non-standard form: $y - 7 = -\\frac{1}{3} (x - \\frac{3}{4})^2$. What is the vertex $(h, k)$?","problem_text":"The equation of a parabola is given in a non-standard form: $y - 7 = -\\frac{1}{3} (x - \\frac{3}{4})^2$. What is the vertex $(h, k)$?","options_latex":["$(-\\frac{1}{3}, -7)$","$(-\\frac{3}{4}, 7)$","$(\\frac{3}{4}, -7)$","$(\\frac{3}{4}, 7)$"],"options_text":["$(-\\frac{1}{3}, -7)$","$(-\\frac{3}{4}, 7)$","$(\\frac{3}{4}, -7)$","$(\\frac{3}{4}, 7)$"],"answer_latex":"(\\frac{3}{4}, 7)","answer_text":"(3/4, 7)","answerIndex":3,"imageUrl":"","imageZoom":"100%","explanation":"The equation $y - 7 = -\\\\frac{1}{3} (x - \\\\frac{3}{4})^2$ is given in the standard vertex form for a vertical parabola, which is $y - k = a(x - h)^2$. The vertex $(h, k)$ is determined by the values of $h$ and $k$. By comparing the given equation to the standard form, we identify the coordinates of the vertex. The $x$-coordinate, $h$, is the value being subtracted from $x$: $x - h = x - \\\\frac{3}{4}$, so $h = \\\\frac{3}{4}$. The $y$-coordinate, $k$, is the value being subtracted from $y$: $y - k = y - 7$, so $k = 7$. Therefore, the vertex $(h, k)$ is $(\\\\frac{3}{4}, 7)$."},{"type":"mcq","problem_latex":"Determine the vertex $(h, k)$ for the quadratic relationship expressed as $10 - y = 4(x - 2)^2 + 5$.","problem_text":"Determine the vertex (h, k) for the quadratic relationship expressed as 10 - y = 4(x - 2)^2 + 5.","options_latex":["(2, 5)","(-2, 5)","(-2, -5)","(2, -5)"],"options_text":["(2, 5)","(-2, 5)","(-2, -5)","(2, -5)"],"answer_latex":"(2, 5)","answer_text":"(2, 5)","answerIndex":0,"explanation":"Rearrange the equation $10 - y = 4(x - 2)^2 + 5$ into the standard vertex form $y = a(x - h)^2 + k$. Subtract $10$ from both sides: $-y = 4(x - 2)^2 + 5 - 10$, which simplifies to $-y = 4(x - 2)^2 - 5$. Multiply the entire equation by $-1$ to isolate $y$: $y = -4(x - 2)^2 + 5$. Comparing this equation to the vertex form $y = a(x - h)^2 + k$, we identify $h$ and $k$. Since $x - h = x - 2$, $h = 2$. Since $k = 5$, the vertex $(h, k)$ is $(2, 5)$."},{"type":"mcq","problem_latex":"The graph of the quadratic function $y=ax^2$ is a vertical stretch (narrower) of the parent function $y=x^2$ if the parameter $a$ satisfies which condition?","problem_text":"The graph of the quadratic function y=ax^2 is a vertical stretch (narrower) of the parent function y=x^2 if the parameter a satisfies which condition?","options_latex":["$a < 0$","$|a| > 1$","$a = 0$","$0 < |a| < 1$"],"options_text":["a < 0","|a| > 1","a = 0","0 < |a| < 1"],"answer_latex":"$|a| > 1$","answer_text":"|a| > 1","answerIndex":1,"explanation":"The function $y=ax^2$ represents a vertical scaling of the parent function $y=x^2$ by a factor of $a$. A vertical stretch, which makes the parabola appear narrower, occurs when the magnitude of the scaling factor $|a|$ is greater than one. If $|a| > 1$, the output $y$-values are $|a|$ \\\\times larger than the $y$-values of $y=x^2$ for the same input $x$. This amplification causes the graph to rise more rapidly, resulting in a vertical stretch or narrowing around the $y$-axis."},{"type":"mcq","problem_latex":"Describe the transformation from the parent function $f(x)=x^2$ to the transformed function $g(x)=\\frac{1}{5}x^2$.","problem_text":"Describe the transformation from the parent function f(x)=x^2 to the transformed function g(x)=(1/5)x^2.","options_latex":["Reflection across the x-axis","Vertical Compression and opens downward","Vertical Stretch and opens upward","Vertical Compression and opens upward"],"options_text":["Reflection across the x-axis","Vertical Compression and opens downward","Vertical Stretch and opens upward","Vertical Compression and opens upward"],"answer_latex":"Vertical Compression and opens upward","answer_text":"Vertical Compression and opens upward","answerIndex":3,"explanation":"The transformed function $g(x)=(1/5)x^2$ is obtained by multiplying the parent function $f(x)=x^2$ by the coefficient $a=1/5$. In transformations of the form $g(x)=af(x)$, when $0 < |a| < 1$, the graph undergoes a vertical compression (widening). Since $0 < 1/5 < 1$, the function is vertically compressed. Because the coefficient $a=1/5$ is positive ($a>0$), the parabola opens upward."},{"type":"mcq","problem_latex":"What specific transformation does the negative sign in the equation $y = -3x^2$ cause compared to the graph of $y=3x^2$?","problem_text":"What specific transformation does the negative sign in the equation y = -3x^2 cause compared to the graph of y=3x^2?","options_latex":["A horizontal shift left 3 units","A vertical shift down 3 units","A reflection across the y-axis","A reflection across the x-axis"],"options_text":["A horizontal shift left 3 units","A vertical shift down 3 units","A reflection across the y-axis","A reflection across the x-axis"],"answer_latex":"A reflection across the x-axis","answer_text":"A reflection across the x-axis","answerIndex":3,"explanation":"The transformation from $y = 3x^2$ to $y = -3x^2$ is an example of $y = -f(x)$, where $f(x) = 3x^2$. Multiplying the entire function output (the $y$-value) by $-1$ negates every $y$-coordinate. This means that a point $(x, y)$ on the original graph becomes $(x, -y)$ on the transformed graph. This specific negation of the $y$-coordinate is defined geometrically as a reflection across the $x$-axis."},{"type":"mcq","problem_latex":"Which function represents a parabola that is wider than $y=x^2$ and opens downward?","problem_text":"Which function represents a parabola that is wider than y=x^2 and opens downward?","options_latex":["$y = -5x^2$","$y = -0.4x^2$","$y = 2x^2$","$y = 0.8x^2$"],"options_text":["y = -5x^2","y = -0.4x^2","y = 2x^2","y = 0.8x^2"],"answer_latex":"$y = -0.4x^2$","answer_text":"y = -0.4x^2","answerIndex":1,"explanation":"The general form of a parabola is $y=ax^2$. For the parabola to open downward, the leading coefficient $a$ must be negative ($a<0$). For the parabola to be wider than $y=x^2$, the absolute value of the coefficient must be less than 1 ($0 < |a| < 1$). In $y = -0.4x^2$, $a = -0.4$. Since $a$ is negative, the parabola opens downward. Since $|a| = |-0.4| = 0.4$, and $0 < 0.4 < 1$, the parabola is wider than $y=x^2$."},{"type":"mcq","problem_latex":"Among the following functions, which graph is the narrowest?","problem_text":"Among the following functions, which graph is the narrowest?","options_latex":["$y = 0.5x^2$","$y = -1.2x^2$","$y = -3.5x^2$","$y = 4x^2$"],"options_text":["y = 0.5x^2","y = -1.2x^2","y = -3.5x^2","y = 4x^2"],"answer_latex":"$y = 4x^2$","answer_text":"y = 4x^2","answerIndex":3,"explanation":"The width of a parabola defined by $y = ax^2$ is controlled by the coefficient $a$. The larger the absolute value of $a$ ($|a|$), the narrower the graph will be because the function is stretched vertically more significantly. Since $y = 4x^2$ has $a=4$, the largest possible positive coefficient, its graph is the narrowest."},{"type":"mcq","problem_latex":"The parabola defined by $f(x) = x^2$ undergoes two transformations: first, a reflection in the x-axis, and second, a vertical stretch by a factor of $3.5$. What is the equation of the resulting function $g(x)$?","problem_text":"The parabola defined by f(x) = x^2 undergoes two transformations: first, a reflection in the x-axis, and second, a vertical stretch by a factor of 3.5. What is the equation of the resulting function g(x)?","options_latex":["$g(x) = -3.5x^2$","$g(x) = 0.35x^2$","$g(x) = -\\frac{1}{3.5}x^2$","$g(x) = 3.5x^2$"],"options_text":["g(x) = -3.5x^2","g(x) = 0.35x^2","g(x) = -(1/3.5)x^2","g(x) = 3.5x^2"],"answer_latex":"$g(x) = -3.5x^2$","answer_text":"g(x) = -3.5x^2","answerIndex":0,"explanation":"The starting function is $f(x) = x^2$. A reflection in the x-axis corresponds to multiplying the function by $-1$, resulting in $h(x) = -x^2$. A vertical stretch by a factor of $k$ is achieved by multiplying the function by $k$. Applying a vertical stretch by a factor of $3.5$ to $h(x)$ yields $g(x) = 3.5 \\\\times h(x)$. Substituting $h(x)$: $g(x) = 3.5(-x^2)$, which simplifies to $g(x) = -3.5x^2$."},{"type":"mcq","problem_latex":"For the function $y = ax^2$, which of the following values of $a$ would result in a parabola that is the narrowest among the options provided, and opens downwards?","problem_text":"For the function y = ax^2, which of the following values of a would result in a parabola that is the narrowest among the options provided, and opens downwards?","options_latex":["$a = -3.8$","$a = -3$","$a = \\frac{7}{2}$","$a = 4$"],"options_text":["a = -3.8","a = -3","a = 7/2 (3.5)","a = 4"],"answer_latex":"$a = -3.8$","answer_text":"a = -3.8","answerIndex":0,"explanation":"For the function $y = ax^2$, the parabola opens downwards only if $a$ is negative ($a < 0$). This requirement is met by $a = -3.8$. The width or narrowness of the parabola is determined by the absolute value of $a$, $|a|$. A larger $|a|$ results in a narrower parabola. Since $a = -3.8$ has the largest absolute magnitude ($|-3.8| = 3.8$) among the provided options, it creates the narrowest parabola that opens downwards."},{"type":"mcq","problem_latex":"A parabola defined by $y = ax^2$ has a range of $y \\le 0$. If this parabola passes through the point $(2, -1)$, determine the exact value of $a$.","problem_text":"A parabola defined by y = ax^2 has a range of y <= 0. If this parabola passes through the point (2, -1), determine the exact value of a.","options_latex":["$a = 1$","$a = -\\frac{1}{4}$","$a = -4$","$a = 4$"],"options_text":["a = 1","a = -1/4","a = -4","a = 4"],"answer_latex":"$a = -\\frac{1}{4}$","answer_text":"a = -1/4","answerIndex":1,"explanation":"The equation $y = ax^2$ defines a parabola with its vertex at $(0, 0)$. Since the range is $y \\\\le 0$, the parabola must open downward, meaning the coefficient $a$ must be negative. To determine the exact value of $a$, substitute the coordinates of the given point $(2, -1)$ into the equation $y = ax^2$: $-1 = a(2)^2$. Simplify the expression: $-1 = 4a$. Solve for $a$ by dividing both sides by 4: $a = -\\\\frac{1}{4}$. This value is negative, which is consistent with the required range."},{"type":"mcq","problem_latex":"Consider two quadratic functions: $f(x) = \\frac{1}{2}x^2$ and $g(x) = -2x^2$. Which statement correctly compares the transformations applied to the base function $y=x^2$ to generate $f(x)$ and $g(x)$?","problem_text":"Consider two quadratic functions: f(x) = (1/2)x^2 and g(x) = -2x^2. Which statement correctly compares the transformations applied to the base function y=x^2 to generate f(x) and g(x)?","options_latex":["Both $f(x)$ and $g(x)$ represent vertical stretches.","$f(x)$ is a vertical compression and $g(x)$ is a vertical stretch and reflection.","$f(x)$ is wider than $g(x)$ but both open upwards.","$f(x)$ is a vertical stretch and $g(x)$ is a vertical compression and reflection."],"options_text":["Both f(x) and g(x) represent vertical stretches.","f(x) is a vertical compression and g(x) is a vertical stretch and reflection.","f(x) is wider than g(x) but both open upwards.","f(x) is a vertical stretch and g(x) is a vertical compression and reflection."],"answer_latex":"$f(x)$ is a vertical compression and $g(x)$ is a vertical stretch and reflection.","answer_text":"f(x) is a vertical compression and g(x) is a vertical stretch and reflection.","answerIndex":1,"explanation":"The transformation of the base function $y=x^2$ to $y=ax^2$ depends on the coefficient $a$. If $0 < |a| < 1$, the function undergoes a vertical compression. If $|a| > 1$, it undergoes a vertical stretch. If $a < 0$, it includes a reflection across the x-axis. For $f(x) = (1/2)x^2$, $a = 1/2$. Since $0 < |1/2| < 1$, $f(x)$ is a vertical compression by a factor of \\\\frac{1}{2}. For $g(x) = -2x^2$, $a = -2$. Since $|-2| = 2 > 1$, $g(x)$ is a vertical stretch by a factor of 2. Since $a$ is negative, $g(x)$ also includes a reflection across the x-axis."},{"type":"mcq","problem_latex":"A new parabola, $h(x) = ax^2$, must be four times wider than the parabola $k(x) = 2x^2$ and must open downwards. What is the required value of $a$?","problem_text":"A new parabola, h(x) = ax^2, must be four times wider than the parabola k(x) = 2x^2 and must open downwards. What is the required value of a?","options_latex":["$a = 8$","$a = -8$","$a = 0.5$","$a = -\\frac{1}{2}$"],"options_text":["a = 8","a = -8","a = 0.5","a = -1/2"],"answer_latex":"$a = -\\frac{1}{2}$","answer_text":"a = -1/2","answerIndex":3,"explanation":"The width of a parabola $y=cx^2$ is determined by the absolute value of the coefficient, $|c|$. To make the new parabola $h(x)$ four \\\\times wider than $k(x) = 2x^2$, the absolute value of the new coefficient $|a|$ must be four \\\\times smaller than the original coefficient $|2|$. We calculate $|a|$: $$|a| = \\\\frac{1}{4} |2| = \\\\frac{2}{4} = \\\\frac{1}{2}$$ Since the parabola must open downwards, the coefficient $a$ must be negative. Using the calculated absolute value, we find the required value of $a$: $$a = -\\\\frac{1}{2}$$"},{"type":"mcq","problem_latex":"Determine the equation of the axis of symmetry for the parabola resulting from shifting the graph of $y = 3(x+5)^2 - 1$ three units to the right and four units up.","problem_text":"Determine the equation of the axis of symmetry for the parabola resulting from shifting the graph of y = 3(x+5)^2 - 1 three units to the right and four units up.","options_latex":["$x = 2$","$x = -5$","$x = 8$","$x = -2$"],"options_text":["x = 2","x = -5","x = 8","x = -2"],"answer_latex":"$x = -2$","answer_text":"x = -2","answerIndex":3,"explanation":"The axis of symmetry (AOS) for a parabola in vertex form $y = a(x-h)^2 + k$ is defined by the equation $x=h$. For the original equation $y = 3(x+5)^2 - 1$, the vertex is $(-5, -1)$, so the original AOS is $x = -5$. Shifting the graph three units to the right means the $x$-coordinate of the vertex, and thus the entire axis of symmetry, increases by 3. Calculate the new AOS: $-5 + 3 = -2$. The vertical shift of four units up does not affect the equation of the vertical axis of symmetry. The equation of the resulting axis of symmetry is $x = -2$."},{"type":"mcq","problem_latex":"What is the axis of symmetry for the function $f(x) = -2(2x - 7)(x + 5)$?","problem_text":"What is the axis of symmetry for the function f(x) = -2(2x - 7)(x + 5)?","options_latex":["$x = \\frac{7}{2}$","$x = -\\frac{3}{4}$","$x = \\frac{1}{2}$","$x = -5$"],"options_text":["x = 7/2","x = -3/4","x = 1/2","x = -5"],"answer_latex":"$x = -\\frac{3}{4}$","answer_text":"x = -3/4","answerIndex":1,"explanation":"The axis of symmetry for a quadratic function in factored form is the midpoint of its roots (x-intercepts). Set $f(x)=0$ to find the roots $x_1$ and $x_2$. From $2x - 7 = 0$, $x_1 = 7/2$. From $x + 5 = 0$, $x_2 = -5$. Calculate the axis of symmetry using the midpoint formula $x = \\\\frac{x_1 + x_2}{2}$. Substitute the roots: $x = \\\\frac{7/2 + (-5)}{2}$. Convert $-5$ to $-10/2$ for common denominators. $x = \\\\frac{7/2 - 10/2}{2}$. Simplify the numerator: $x = \\\\frac{-3/2}{2}$. Therefore, $x = -3/4$."},{"type":"mcq","problem_latex":"If the quadratic function $g(x) = 4x^2 - kx + 10$ has an axis of symmetry defined by $x = 3.5$, find the value of the coefficient $k$.","problem_text":"If the quadratic function g(x) = 4x^2 - kx + 10 has an axis of symmetry defined by x = 3.5, find the value of the coefficient k.","options_latex":["$k = 14$","$k = -28$","$k = 28$","$k = 7$"],"options_text":["k = 14","k = -28","k = 28","k = 7"],"answer_latex":"$k = 28$","answer_text":"k = 28","answerIndex":2,"explanation":"The axis of symmetry for a quadratic function $g(x) = ax^2 + bx + c$ is found using the formula $x = \\\\frac{-b}{2a}$. For the given function $g(x) = 4x^2 - kx + 10$, we identify $a=4$ and $b=-k$. Since the axis of symmetry is $x=3.5$, we substitute these values into the formula: $3.5 = \\\\frac{-(-k)}{2(4)}$. Simplify the expression: $3.5 = \\\\frac{k}{8}$. To solve for $k$, multiply both sides by 8: $k = 3.5 \\\\times 8$. Therefore, $k = 28$."},{"type":"mcq","problem_latex":"Find the equation for the axis of symmetry of the parabola defined by $y = 5x^2 + 15x - 2$.","problem_text":"Find the equation for the axis of symmetry of the parabola defined by y = 5x^2 + 15x - 2.","options_latex":["$x = -\\frac{3}{2}$","$x = 1.5$","$x = -3$","$x = \\frac{1}{5}$"],"options_text":["x = -3/2","x = 1.5","x = -3","x = 1/5"],"answer_latex":"$x = -\\frac{3}{2}$","answer_text":"x = -3/2","answerIndex":0,"explanation":"The equation for the axis of symmetry of a parabola defined by the standard quadratic form $y = ax^2 + bx + c$ is $x = -\\\\frac{b}{2a}$. From the given equation $y = 5x^2 + 15x - 2$, we identify the coefficients $a=5$ and $b=15$. Substitute these values into the axis of symmetry formula: $x = -\\\\frac{15}{2(5)}$. Simplify the denominator: $x = -\\\\frac{15}{10}$. Reduce the fraction by dividing both the numerator and denominator by the greatest common factor, 5: $x = -\\\\frac{3}{2}$."},{"type":"mcq","problem_latex":"For the quadratic function $h(x) = \\frac{1}{3}x^2 - 4x + 1$, what is the equation of the axis of symmetry?","problem_text":"For the quadratic function h(x) = (1/3)x^2 - 4x + 1, what is the equation of the axis of symmetry?","options_latex":["$x = 6$","$x = 12$","$x = 2$","$x = -6$"],"options_text":["x = 6","x = 12","x = 2","x = -6"],"answer_latex":"$x = 6$","answer_text":"x = 6","answerIndex":0,"explanation":"The axis of symmetry for a quadratic function $h(x) = ax^2 + bx + c$ is found using the formula $x = -\\\\frac{b}{2a}$. For the given function $h(x) = \\\\frac{1}{3}x^2 - 4x + 1$, we identify $a = \\\\frac{1}{3}$ and $b = -4$. Substitute these values into the formula: $x = -\\\\frac{(-4)}{2(\\\\frac{1}{3})}$. This simplifies to $x = \\\\frac{4}{\\\\frac{2}{3}}$. Dividing by a fraction is equivalent to multiplying by its reciprocal: $x = 4 \\\\times \\\\frac{3}{2} = \\\\frac{12}{2} = 6$. The equation of the axis of symmetry is $x = 6$."},{"type":"mcq","problem_latex":"A parabola has a vertex $(4, 1)$ and passes through $(6, 4)$. What is its equation?","problem_text":"A parabola has a vertex $(4, 1)$ and passes through $(6, 4)$. What is its equation?","options_latex":["y = $\\frac{3}{4}(x - 4)^2 + 1$","y = $\\frac{3}{4}(x + 4)^2 + 1$","y = $\\frac{4}{3}(x - 4)^2 + 1$","y = $3(x - 4)^2 + 1$"],"options_text":["y = $\\frac{3}{4}(x - 4)^2 + 1$","y = $\\frac{3}{4}(x + 4)^2 + 1$","y = $\\frac{4}{3}(x - 4)^2 + 1$","y = $3(x - 4)^2 + 1$"],"answer_latex":"y = \\frac{3}{4}(x - 4)^2 + 1","answer_text":"y = 3/4(x - 4)^2 + 1","answerIndex":0,"imageUrl":"","imageZoom":"100%","explanation":"Explanation could not be generated."},{"type":"mcq","problem_latex":"Find the equation of the parabola if the vertex is $(1, 7)$ and it contains the point $(-1, 5)$.","problem_text":"Find the equation of the parabola if the vertex is $(1, 7)$ and it contains the point $(-1, 5)$.","options_latex":["y = $-\\frac{1}{4}(x - 1)^2 + 7$","y = $\\frac{1}{2}(x - 1)^2 + 7$","y = $-\\frac{1}{2}(x - 1)^2 + 7$","y = $-2(x - 1)^2 + 7$"],"options_text":["y = $-\\frac{1}{4}(x - 1)^2 + 7$","y = $\\frac{1}{2}(x - 1)^2 + 7$","y = $-\\frac{1}{2}(x - 1)^2 + 7$","y = $-2(x - 1)^2 + 7$"],"answer_latex":"y = -\\frac{1}{2}(x - 1)^2 + 7","answer_text":"y = -1/2(x - 1)^2 + 7","answerIndex":2,"imageUrl":"","imageZoom":"100%","explanation":"The vertex form of a parabola is $y = a(x - h)^2 + k$. First, substitute the vertex $(h, k) = (1, 7)$ into the equation: $y = a(x - 1)^2 + 7$. Next, use the given point $(-1, 5)$ to solve for $a$. Substitute $x = -1$ and $y = 5$: $5 = a(-1 - 1)^2 + 7$. Simplify the expression inside the parentheses: $5 = a(-2)^2 + 7$. Square the term: $5 = 4a + 7$. Subtract 7 from both sides: $-2 = 4a$. Divide by 4 to find $a$: $a = -\\\\frac{2}{4} = -\\\\frac{1}{2}$. Substitute $a = -\\\\frac{1}{2}$ back into the vertex form to get the final equation: $y = -\\\\frac{1}{2}(x - 1)^2 + 7$."},{"type":"mcq","problem_latex":"Find the equation of the parabola, in vertex form $y = a(x - h)^2 + k$, with vertex $(3, 1)$ that passes through the point $(1, 5)$.","problem_text":"Find the equation of the parabola, in vertex form y = a(x - h)^2 + k, with vertex (3, 1) that passes through the point (1, 5).","options_latex":["$y = -\\frac{1}{2}(x - 3)^2 + 1$","$y = 2(x - 3)^2 + 1$","$y = (x + 3)^2 + 1$","$y = (x - 3)^2 + 1$"],"options_text":["y = -1/2(x - 3)^2 + 1","y = 2(x - 3)^2 + 1","y = (x + 3)^2 + 1","y = (x - 3)^2 + 1"],"answer_latex":"$y = (x - 3)^2 + 1$","answer_text":"y = (x - 3)^2 + 1","answerIndex":3,"explanation":"The vertex form of a parabola is $y = a(x - h)^2 + k$. Given the vertex $(h, k) = (3, 1)$, substitute these values into the equation: $y = a(x - 3)^2 + 1$. To find the stretch factor $a$, substitute the coordinates of the given point $(1, 5)$ into the current equation, using $x=1$ and $y=5$: $5 = a(1 - 3)^2 + 1$. Simplify the expression inside the parenthesis: $5 = a(-2)^2 + 1$. Square the term: $5 = 4a + 1$. Subtract 1 from both sides: $4 = 4a$. Divide by 4: $a = 1$. Substitute $a=1$ back into the vertex form: $y = 1(x - 3)^2 + 1$, which simplifies to $y = (x - 3)^2 + 1$."},{"type":"mcq","problem_latex":"Write the vertex form equation for a parabola having a vertex at $(-2, -5)$ and passing through the point $(-1, -3)$.","problem_text":"Write the vertex form equation for a parabola having a vertex at (-2, -5) and passing through the point (-1, -3).","options_latex":["$y = 2(x + 2)^2 - 5$","$y = 2(x - 2)^2 - 5$","$y = \\frac{1}{2}(x + 2)^2 - 5$","$y = -2(x + 2)^2 - 5$"],"options_text":["y = 2(x + 2)^2 - 5","y = 2(x - 2)^2 - 5","y = 1/2(x + 2)^2 - 5","y = -2(x + 2)^2 - 5"],"answer_latex":"$y = 2(x + 2)^2 - 5$","answer_text":"y = 2(x + 2)^2 - 5","answerIndex":0,"explanation":"The vertex form of a parabola is $y = a(x - h)^2 + k$. Substitute the vertex $(h, k) = (-2, -5)$ into the formula: $y = a(x - (-2))^2 + (-5)$, which simplifies to $y = a(x + 2)^2 - 5$. To find the vertical stretch factor $a$, substitute the passing point $(x, y) = (-1, -3)$ into the current equation: $-3 = a(-1 + 2)^2 - 5$. Simplify the expression: $-3 = a(1)^2 - 5$. Solving for $a$: $-3 = a - 5$, so $a = 2$. Substitute $a = 2$ back into the simplified vertex form to obtain the final equation: $y = 2(x + 2)^2 - 5$."},{"type":"mcq","problem_latex":"A quadratic function has a vertex at $(4, 0)$ and contains the point $(0, 8)$. Which equation represents this function?","problem_text":"A quadratic function has a vertex at (4, 0) and contains the point (0, 8). Which equation represents this function?","options_latex":["$y = 2(x - 4)^2$","$y = \\frac{1}{2}(x + 4)^2$","$y = \\frac{1}{2}(x - 4)^2$","$y = -\\frac{1}{2}(x - 4)^2$"],"options_text":["y = 2(x - 4)^2","y = 1/2(x + 4)^2","y = 1/2(x - 4)^2","y = -1/2(x - 4)^2"],"answer_latex":"$y = \\frac{1}{2}(x - 4)^2$","answer_text":"y = 1/2(x - 4)^2","answerIndex":2,"explanation":"Begin by using the vertex form of a quadratic function, $y = a(x - h)^2 + k$. Substitute the vertex $(h, k) = (4, 0)$ into the equation: $y = a(x - 4)^2 + 0$, which simplifies to $y = a(x - 4)^2$. Now, use the given point $(0, 8)$ to solve for the stretch factor $a$. Substitute $x = 0$ and $y = 8$: $8 = a(0 - 4)^2$. Calculate the expression: $8 = a(-4)^2$, so $8 = 16a$. Solve for $a$ by dividing both sides by 16: $a = \\\\frac{8}{16} = \\\\frac{1}{2}$. Substitute $a = \\\\frac{1}{2}$ back into the simplified vertex form to get the final equation: $y = \\\\frac{1}{2}(x - 4)^2$."},{"type":"mcq","problem_latex":"Determine the equation in vertex form for the parabola whose vertex is $(1, 7)$ and which passes through the point $(2, 4)$.","problem_text":"Determine the equation in vertex form for the parabola whose vertex is (1, 7) and which passes through the point (2, 4).","options_latex":["$y = -3(x - 1)^2 + 7$","$y = 3(x + 1)^2 + 7$","$y = -3(x + 1)^2 + 7$","$y = -3(x - 1)^2 - 7$"],"options_text":["y = -3(x - 1)^2 + 7","y = 3(x + 1)^2 + 7","y = -3(x + 1)^2 + 7","y = -3(x - 1)^2 - 7"],"answer_latex":"$y = -3(x - 1)^2 + 7$","answer_text":"y = -3(x - 1)^2 + 7","answerIndex":0,"explanation":"The vertex form of a parabola is $y = a(x - h)^2 + k$. Substitute the given vertex $(h, k) = (1, 7)$ into the equation: $y = a(x - 1)^2 + 7$. Use the given point $(2, 4)$ to determine the value of $a$. Substitute $x=2$ and $y=4$: $4 = a(2 - 1)^2 + 7$. Simplify the expression: $4 = a(1)^2 + 7$, so $4 = a + 7$. Solving for $a$ gives $a = 4 - 7$, or $a = -3$. Substitute $a = -3$ back into the vertex form to obtain the final equation: $y = -3(x - 1)^2 + 7$."},{"type":"mcq","problem_latex":"If the vertex of a quadratic function is located at $(-1, 6)$ and the graph passes through the point $(1, -2)$, what is the function's equation?","problem_text":"If the vertex of a quadratic function is located at (-1, 6) and the graph passes through the point (1, -2), what is the function's equation?","options_latex":["$y = 2(x + 1)^2 + 6$","$y = -2(x + 1)^2 + 6$","$y = -\\frac{1}{2}(x + 1)^2 + 6$","$y = -2(x - 1)^2 + 6$"],"options_text":["y = 2(x + 1)^2 + 6","y = -2(x + 1)^2 + 6","y = -1/2(x + 1)^2 + 6","y = -2(x - 1)^2 + 6"],"answer_latex":"$y = -2(x + 1)^2 + 6$","answer_text":"y = -2(x + 1)^2 + 6","answerIndex":1,"explanation":"The equation for a quadratic function in vertex form is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex. Substitute the given vertex $(-1, 6)$, so $h = -1$ and $k = 6$: $y = a(x - (-1))^2 + 6$, which simplifies to $y = a(x + 1)^2 + 6$. To find the stretch factor $a$, substitute the coordinates of the second point $(1, -2)$ into the equation: $-2 = a(1 + 1)^2 + 6$. Simplify the expression: $-2 = a(2)^2 + 6$, or $-2 = 4a + 6$. Subtract 6 from both sides: $-8 = 4a$. Divide by 4: $a = \\\\frac{-8}{4} = -2$. Substitute $a = -2$ back into the vertex form to obtain the final equation: $y = -2(x + 1)^2 + 6$."},{"type":"mcq","problem_latex":"Determine the $y$-intercept of the parabola defined by the equation $y = 2(x - 3)^2 - 8$.","problem_text":"Determine the y-intercept of the parabola defined by the equation y = 2(x - 3)^2 - 8.","options_latex":["$(0, -2)$","$(0, -8)$","$(0, 18)$","$(0, 10)$"],"options_text":["(0, -2)","(0, -8)","(0, 18)","(0, 10)"],"answer_latex":"$(0, 10)$","answer_text":"(0, 10)","answerIndex":3,"explanation":"To determine the y-intercept, set $x=0$ in the equation $y = 2(x - 3)^2 - 8$. Substitute $x=0$: $y = 2(0 - 3)^2 - 8$. Simplify the expression inside the parentheses: $y = 2(-3)^2 - 8$. Square the term: $y = 2(9) - 8$. Perform multiplication and subtraction: $y = 18 - 8 = 10$. The y-intercept is $(0, 10)$."},{"type":"mcq","problem_latex":"Find the $x$-intercepts of the function $y = (x + 1)^2 - 9$.","problem_text":"Find the x-intercepts of the function y = (x + 1)^2 - 9.","options_latex":["$(2, 0)$ and $(-4, 0)$","$(-1, 0)$ and $(9, 0)$","$(4, 0)$ and $(-2, 0)$","$(-1, 0)$ and $(-9, 0)$"],"options_text":["(2, 0) and (-4, 0)","(-1, 0) and (9, 0)","(4, 0) and (-2, 0)","(-1, 0) and (-9, 0)"],"answer_latex":"$(2, 0)$ and $(-4, 0)$","answer_text":"(2, 0) and (-4, 0)","answerIndex":0,"explanation":"To find the x-intercepts, set $y=0$. The equation is $0 = (x + 1)^2 - 9$. Isolate the squared term by adding 9 to both sides: $9 = (x + 1)^2$. Take the square root of both sides, remembering both positive and negative roots: $\\\\pm \\\\sqrt{9} = x + 1$. This simplifies to $\\\\pm 3 = x + 1$. Solve the two resulting cases. Case 1: $3 = x + 1$, yielding $x = 2$. Case 2: $-3 = x + 1$, yielding $x = -4$. The x-intercepts are $(2, 0)$ and $(-4, 0)$."},{"type":"mcq","problem_latex":"What are the $x$-intercepts and the $y$-intercept for the parabola $y = -3(x - 1)^2 + 12$?","problem_text":"What are the x-intercepts and the y-intercept for the parabola y = -3(x - 1)^2 + 12?","options_latex":["X-intercepts: $(3, 0), (-1, 0)$; Y-intercept: $(0, 11)$","X-intercepts: $(2, 0), (0, 0)$; Y-intercept: $(0, 9)$","X-intercepts: $(3, 0), (-1, 0)$; Y-intercept: $(0, 9)$","X-intercepts: $(1, 0), (-2, 0)$; Y-intercept: $(0, 12)$"],"options_text":["X-intercepts: (3, 0), (-1, 0); Y-intercept: (0, 11)","X-intercepts: (2, 0), (0, 0); Y-intercept: (0, 9)","X-intercepts: (3, 0), (-1, 0); Y-intercept: (0, 9)","X-intercepts: (1, 0), (-2, 0); Y-intercept: (0, 12)"],"answer_latex":"X-intercepts: $(3, 0), (-1, 0)$; Y-intercept: $(0, 9)$","answer_text":"X-intercepts: (3, 0), (-1, 0); Y-intercept: (0, 9)","answerIndex":2,"explanation":"To find the y-intercept, set $x=0$: $y = -3(0 - 1)^2 + 12$. Simplifying gives $y = -3(-1)^2 + 12 = -3(1) + 12 = 9$. The y-intercept is $(0, 9)$. To find the x-intercepts, set $y=0$: $0 = -3(x - 1)^2 + 12$. Isolate the squared term: $3(x - 1)^2 = 12$. Divide by 3: $(x - 1)^2 = 4$. Take the square root of both sides: $x - 1 = \\\\pm \\\\sqrt{4}$, so $x - 1 = \\\\pm 2$. Case 1: $x - 1 = 2$, so $x = 3$. Case 2: $x - 1 = -2$, so $x = -1$. The x-intercepts are $(3, 0)$ and $(-1, 0)$."},{"type":"mcq","problem_latex":"Identify the $x$-intercept(s) of the function $y = 4(x + 5)^2$.","problem_text":"Identify the x-intercept(s) of the function y = 4(x + 5)^2.","options_latex":["$(-5, 0)$","No $x$-intercepts","$(-5, 0)$ and $(5, 0)$","$(5, 0)$"],"options_text":["(-5, 0)","No x-intercepts","(-5, 0) and (5, 0)","(5, 0)"],"answer_latex":"$(-5, 0)$","answer_text":"(-5, 0)","answerIndex":2,"explanation":"To find the x-intercept(s), set the function $y$ equal to zero: $0 = 4(x + 5)^2$. Divide both sides by 4: $0 = (x + 5)^2$. Take the square root of both sides: $\\\\sqrt{0} = \\\\sqrt{(x + 5)^2}$, which simplifies to $0 = x + 5$. Subtract 5 from both sides to solve for $x$: $x = -5$. The function $y = 4(x + 5)^2$ has only one x-intercept at $(-5, 0)$."},{"type":"mcq","problem_latex":"Determine the $y$-intercept of the quadratic equation $y = -(x - 2)^2 - 5$.","problem_text":"Determine the y-intercept of the quadratic equation y = -(x - 2)^2 - 5.","options_latex":["$(0, -5)$","$(0, -7)$","$(0, -9)$","$(0, 4)$"],"options_text":["(0, -5)","(0, -7)","(0, -9)","(0, 4)"],"answer_latex":"$(0, -9)$","answer_text":"(0, -9)","answerIndex":2,"explanation":"To determine the y-intercept, set $x=0$ in the quadratic equation $y = -(x - 2)^2 - 5$. Substitute $x=0$: $y = -(0 - 2)^2 - 5$. Simplify the terms within the parentheses: $y = -(-2)^2 - 5$. Square the term: $y = -(4) - 5$. Calculate the final value: $y = -4 - 5 = -9$. The y-intercept is the point $(0, -9)$."},{"type":"mcq","problem_latex":"What are the coordinates of the turning point (vertex) for the quadratic function $y = 5(x - 2)^2 + 7$?","problem_text":"What are the coordinates of the turning point (vertex) for the quadratic function y = 5(x - 2)^2 + 7?","options_latex":["$(-2, 7)$","$(-5, -7)$","$(2, 7)$","$(2, -7)$"],"options_text":["(-2, 7)","(-5, -7)","(2, 7)","(2, -7)"],"answer_latex":"$(2, 7)$","answer_text":"(2, 7)","answerIndex":2,"explanation":"The quadratic function is in vertex form, $y = a(x - h)^2 + k$. In this form, the coordinates of the turning point (vertex) are $(h, k)$. Comparing the given function $y = 5(x - 2)^2 + 7$ to the vertex form, we identify $h=2$ and $k=7$. Therefore, the vertex is $(2, 7)$."},{"type":"mcq","problem_latex":"Consider the parent function $y = x^2$. Which of the following functions represents a reflection across the x-axis?","problem_text":"Consider the parent function y = x^2. Which of the following functions represents a reflection across the x-axis?","options_latex":["$y = (x + 7)^2 - 1$","$y = 2(x + 1)^2 - 5$","$y = 0.5(x - 3)^2 + 1$","$y = -3(x - 4)^2 + 2$"],"options_text":["y = (x + 7)^2 - 1","y = 2(x + 1)^2 - 5","y = 0.5(x - 3)^2 + 1","y = -3(x - 4)^2 + 2"],"answer_latex":"$y = -3(x - 4)^2 + 2$","answer_text":"y = -3(x - 4)^2 + 2","answerIndex":3,"explanation":"The transformation required for a reflection across the x-axis is $y = -f(x)$. For a quadratic function written in vertex form, $y = a(x - h)^2 + k$, a reflection across the x-axis is represented by a negative value for the leading coefficient $a$. The parent function $y = x^2$ has a positive coefficient ($a=1$) and opens upward. The given function $y = -3(x - 4)^2 + 2$ has $a = -3$. Since $a$ is negative, the parabola opens downward, which is the definition of a reflection across the x-axis."},{"type":"mcq","problem_latex":"Compared to the parabola $y_B = 0.5(x - 1)^2 + 5$, how is the parabola $y_A = 4(x - 1)^2 + 5$ transformed?","problem_text":"Compared to the parabola y_B = 0.5(x - 1)^2 + 5, how is the parabola y_A = 4(x - 1)^2 + 5 transformed?","options_latex":["It is shifted 3.5 units up.","It is vertically stretched, making it narrower.","It is vertically compressed, making it wider.","It is reflected across the x-axis."],"options_text":["It is shifted 3.5 units up.","It is vertically stretched, making it narrower.","It is vertically compressed, making it wider.","It is reflected across the x-axis."],"answer_latex":"It is vertically stretched, making it narrower.","answer_text":"It is vertically stretched, making it narrower.","answerIndex":1,"explanation":"The parabolas are in vertex form $y = a(x - h)^2 + k$. Both $y_A$ and $y_B$ share the same vertex $(h, k) = (1, 5)$, meaning there is no horizontal or vertical shift. The transformation is determined by the coefficient $a$. For $y_B$, $a_B = 0.5$. For $y_A$, $a_A = 4$. Since $|a_A| = 4$ is greater than $|a_B| = 0.5$, the parabola $y_A$ is vertically stretched compared to $y_B$. A larger absolute value of $a$ results in a narrower graph."},{"type":"mcq","problem_latex":"What is the equation for the axis of symmetry for the quadratic function $y = -2(x + 5)^2 - 8$?","problem_text":"What is the equation for the axis of symmetry for the quadratic function y = -2(x + 5)^2 - 8?","options_latex":["$x = -5$","$y = 8$","$x = 5$","$y = -8$"],"options_text":["x = -5","y = 8","x = 5","y = -8"],"answer_latex":"$x = -5$","answer_text":"x = -5","answerIndex":0,"explanation":"The quadratic function is in vertex form, $y = a(x - h)^2 + k$. The axis of symmetry is always the vertical line $x = h$. In the given function, $y = -2(x + 5)^2 - 8$, we compare this to the standard form. Since $(x + 5)$ is equivalent to $(x - (-5))$, we determine that $h = -5$. Therefore, the equation for the axis of symmetry is $x = -5$."},{"type":"mcq","problem_latex":"Which quadratic function definitely has no x-intercepts?","problem_text":"Which quadratic function definitely has no x-intercepts?","options_latex":["$y = -1(x + 3)^2 + 1$","$y = -5(x + 2)^2 + 9$","$y = 3(x - 1)^2 - 4$","$y = 4(x - 5)^2 + 6$"],"options_text":["y = -1(x + 3)^2 + 1","y = -5(x + 2)^2 + 9","y = 3(x - 1)^2 - 4","y = 4(x - 5)^2 + 6"],"answer_latex":"$y = 4(x - 5)^2 + 6$","answer_text":"y = 4(x - 5)^2 + 6","answerIndex":3,"explanation":"The quadratic function $y = 4(x - 5)^2 + 6$ is in vertex form $y = a(x - h)^2 + k$. The vertex is $(5, 6)$, and the parabola opens upward because the leading coefficient $a=4$ is positive. Since $(x - 5)^2 \\\\ge 0$ for all real $x$, the smallest possible value for the function is $y = 4(0) + 6 = 6$. Because the minimum $y$-value is $6$, which is greater than $0$, the graph never crosses the x-axis, meaning there are no x-intercepts."}]