CTJan27 Online JMSS Exam Review - Quadratics and Logs

A parabola is defined by the equation $y = -2x^2 + 12x - 11$. What is the equation of its axis of symmetry?

A quadratic function defines a parabola with x-intercepts at $(-5, 0)$ and $(3, 0)$. The parabola also passes through the point $(1, -24)$. Which of the following is the equation of this parabola?

The vertex of a parabola is at $(-4, 7)$ and it opens downwards. Which of the following equations could represent this parabola?

For what value of the constant $k$ will the parabola given by the equation $y = x^2 - 6x + k$ have exactly one x-intercept (i.e., its vertex touches the x-axis)?

Consider two parabolas, $P_1$ with equation $y = x^2 - 8x + 15$ and $P_2$ with equation $y = -2(x-4)^2 + 5$. Which of the following statements is true?

Consider the quadratic function $y = -x^2 + 4x - 1$. A table of values is partially completed below. By analyzing the symmetry of the parabola, what are the coordinates of the vertex?

A quadratic function is represented by the table of values below. Which equation correctly models this function?

The table below represents points on a parabola. What are the axis of symmetry and the y-intercept of the parabola?

The table below shows some points for the quadratic function $y = x^2 - 6x + 8$. Based on the pattern of symmetry, which of the following points would also be on the graph of this function?

The table below shows points for a quadratic function of the form $y=ax^2+bx+c$. By analyzing the second differences in the y-values, determine the equation of the parabola.

The parabola represented by the equation $y = -\frac{1}{2}(x+4)^2 - 6$ has its vertex at which point and opens in which direction?

Which of the following is the vertex form of the quadratic equation $y = 2x^2 - 12x + 11$?

A parabola has its vertex at $(3, -5)$ and passes through the point $(1, 7)$. What is the equation of the parabola in vertex form?

The height $H$ (in meters) of a diver above the water, $t$ seconds after diving from a platform, is modeled by the equation $H(t) = -5(t-0.8)^2 + 13$. What is the maximum height reached by the diver and when does it occur?

The graph of $y = x^2$ is transformed to obtain the graph of $y = -\frac{1}{3}(x-5)^2 + 2$. Which of the following statements accurately describes the transformations?

The parent function $f(x) = x^2$ is reflected across the x-axis, then translated 3 units to the left and 5 units down. What is the equation of the transformed function, $g(x)$?

A parabola has its vertex at $(-2, 7)$ and passes through the point $(0, -5)$. Which of the following is the equation of the parabola?

Which of the following descriptions accurately represents the transformations applied to the parent function $y=x^2$ to obtain the graph of $y = -2x^2 + 8x - 5$?

The vertex of the parabola $f(x) = (x-4)^2 + 1$ is at $(4, 1)$. If the function is transformed to $g(x) = -2(x+1)^2 - 3$, what is the translation vector that maps the vertex of $f(x)$ to the vertex of $g(x)$?

Consider the function $f(x) = 2(x-1)^2 + 5$. A new function $g(x)$ is created by applying the following sequence of transformations to $f(x)$: a translation 3 units to the left, followed by a reflection across the line $y=5$. What is the equation of $g(x)$?

A parabola is defined by the equation $y = x^2 - 4x + 5$ and a line is defined by $y = -2x + k$. If the graphs of these two equations are drawn on the same set of axes, for which value of $k$ will there be exactly one point of intersection?

The graph of the function $y = 2x^2 + 5x - 3$ is created. The solutions to an equation are found by identifying the x-intercepts of this graph, which are determined to be $x = -3$ and $x = 0.5$. Which of the following equations was being solved?

Consider the graphs of the functions $y = x^2$ and $y = \frac{8}{x}$. If both are drawn on the same coordinate plane, how many real solutions does the equation $x^2 = \frac{8}{x}$ have, and what is the x-coordinate of the intersection point(s)?

The graph of $y = -(x-3)^2 + 1$ is a parabola. The solutions to the equation $-(x-3)^2 + 1 = k$ correspond to the x-coordinates of the intersection points between the parabola and the horizontal line $y=k$. For which value of $k$ does the equation have no real solutions?

The graphs of $f(x) = x^2 - 2x - 3$ and $g(x) = x - 3$ are plotted. To solve the inequality $x^2 - 2x - 3 > x - 3$, we look for the x-values where the graph of $f(x)$ is above the graph of $g(x)$. The graphs intersect at the points where $x=0$ and $x=3$. What is the solution set for the inequality?

The area of a rectangular garden is represented by the trinomial $x^2 + 2x - 143$. If the length and width are binomials with integer coefficients, what are the dimensions of the garden?

Completely factor the trinomial $8x^2 - 38xy + 35y^2$.

A quadratic trinomial is given as $15x^2 + kx - 18$. If one of its factors is $(5x - 3)$, what is the value of the coefficient $k$?

The first three terms of a quadratic sequence are given by substituting $n=1, 2, 3$ into the expression $T_n = n^2 + 14n + 48$. For what value of $n$ will the term value $T_n$ be equal to zero?

The expression $49x^4 - 70x^2y^3 + ky^6$ is a perfect square trinomial. What must be the value of $k$?

Factor the expression $3x^3 - 75x$ completely.

Which of the following represents the complete factorization of $x^4 - 81$?

Factor the expression $100a^2 - \frac{1}{49}b^2$ completely.

Factor the expression $(3x+2)^2 - 25$ completely.

Factor the expression $16(2x-1)^2 - 81y^2$ completely.

Solve the quadratic equation for $x$: $6x^2 - 11x - 10 = 0$.

Find the solution set for the equation $2x^2 = 3x + 20$.

Solve for $x$ by factoring the difference of squares: $(2x - 1)^2 - 49 = 0$.

What is the solution to the equation $25x^2 + 9 = -30x$?

The product of two consecutive positive odd integers is 143. What is the value of the larger integer?

Solve the quadratic equation $3x^2 + 5x - 7 = 0$ by completing the square.

The quadratic equation $x^2 - 2kx + 3k = 0$ is solved for $x$ by completing the square. Which of the following represents the solution for $x$ in terms of $k$, assuming real solutions exist?

The area of a rectangular garden is 91 square meters. The length is 6 meters longer than the width. Find the width of the garden by setting up a quadratic equation and solving it by completing the square.

A student's work for solving $2x^2 - 8x - 5 = 0$ by completing the square is shown below. In which step was the first mistake made?

• Step 1: $2x^2 - 8x = 5$
• Step 2: $x^2 - 4x = \frac{5}{2}$
• Step 3: $x^2 - 4x + 16 = \frac{5}{2} + 16$
• Step 4: $(x - 4)^2 = \frac{37}{2}$

Solve the equation $\frac{1}{x} + \frac{1}{x+2} = \frac{1}{3}$ by first transforming it into a standard quadratic equation and then using the method of completing the square. (Assume $x eq 0, -2$)

To solve the quadratic equation $3x^2 - 5 = -7x$ using the quadratic formula, it must first be rewritten in the standard form $ax^2 + bx + c = 0$. What are the solutions for $x$?

For what value of the constant $k$ will the quadratic equation $x^2 - 6x + k = 0$ have exactly one distinct real root?

The product of two consecutive positive odd integers is 143. Which of the following equations models this situation, and what is the larger of the two integers? Let $n$ be the smaller integer.

Find all real solutions to the equation $x - \frac{4}{x} = 3$.

A quadratic equation is given by $2x^2 - 8x + 3 = 0$. The roots of this equation can be found using the quadratic formula. Without solving for the specific roots, what is the product of the roots?

For what range of values of the constant $k$ does the quadratic equation $x^2 - kx + (k+3) = 0$ have two distinct real roots?

The quadratic equation $(p+1)x^2 + 2(p+3)x + (p+8) = 0$ has exactly one real root (a repeated root). Given that $p eq -1$, what is the value of $p$?

Suppose the quadratic equation $ax^2 + bx + c = 0$ has no real roots, where $a$ and $c$ are positive real numbers. What can be definitively concluded about the number of real roots of the equation $ax^2 + bx - c = 0$?

Consider the equation $2x^2 + (k-1)x + (k+1) = 0$. For which of the following values of $k$ does the equation have no real roots?

A quadratic equation $x^2 + px + q = 0$ has two distinct real roots. If the sum of the roots is equal to their product, which of the following conditions must be true?

A theatre company finds that if they price their tickets at 10 dollars, they sell 200 tickets. For every 1 dollar increase in price, they sell 5 fewer tickets. What ticket price will maximize their total revenue?

A farmer has 120 meters of fencing to create a rectangular garden. What is the maximum possible area of the garden?

The quadratic function $f(x) = x^2 - 4kx + 5k^2 - 2k + 1$ has a minimum value that depends on the parameter $k$. What is this minimum value expressed in terms of $k$?

The height of a projectile after $t$ seconds is given by $h(t) = 80t - 5t^2$. A tracking device gives a reading $V$ related to the height by the formula $V = 1600 - \frac{h(t)}{2}$. To get the minimum reading for $V$, what must the height of the projectile be?

What is the maximum value of the function $f(x) = -x^2 + 6x - 5$ on the closed interval $[0, 4]$?

A farmer wants to build a rectangular pen using 100 meters of fencing. The pen is built against a long, straight wall, so fencing is only needed for the three other sides. Which quadratic function models the area, $A$, of the pen in terms of its width, $x$, (the side perpendicular to the wall)?

A diver jumps from a platform 10 meters above the water. Her height, $h$ (in meters), above the water after $t$ seconds is modeled by the function $h(t) = -5t^2 + 5t + 10$. A second diver jumps from a lower platform 8 meters high with an initial upward velocity of 6 m/s. Assuming the same gravitational acceleration (a coefficient of $-5$ for the $t^2$ term), which function correctly models the second diver's height?

The number of diagonals, $d$, in a polygon with $n$ sides follows a quadratic pattern. A triangle ($n=3$) has 0 diagonals, a quadrilateral ($n=4$) has 2 diagonals, and a pentagon ($n=5$) has 5 diagonals. Which quadratic function models the number of diagonals for a polygon with $n$ sides?

A concert promoter sells tickets for 50 dollars each and expects to sell 1200 tickets. Market research suggests that for each 5 dollar increase in ticket price, 100 fewer tickets will be sold. Which quadratic function models the total revenue, $R$, as a function of $x$, where $x$ represents the number of 5 dollar price increases?

The number of dots in a sequence of shapes follows a quadratic pattern. The first shape has 1 dot, the second has 5 dots, the third has 13 dots, and the fourth has 25 dots. Which function $D(n)$ models the number of dots in the $n$-th shape?

If the exponential equation $3^{y+1} = x$ is rewritten in logarithmic form, which of the following expressions is correct?

The logarithmic statement $\log_{1/2}(16) = -4$ is the inverse of which exponential equation?

To solve for $x$ in the equation $5^x = \frac{1}{25}$, you can use logarithms. What is the value of $x$, which is equivalent to finding $\log_5(\frac{1}{25})$?

The function $f(x) = 7^x$ is an exponential function. Since logarithmic functions are the inverses of exponential functions, what is the correct inverse function, $f^{-1}(x)$?

By converting the logarithmic equation to its equivalent exponential form, find the value of the base $b$ in the equation $\log_b(64) = 3$.

Which of the following is the correct logarithmic form of the equation $8^{1/3} = 2$?

The statement $\log_k(m) = n$ is equivalent to which of the following exponential statements, assuming $k > 0$ and $k eq 1$?

If $5^{-2} = \frac{1}{25}$, which of the following logarithmic expressions is true?

What is the value of $x$ if $\log_x(81) = 4$?

In the context of logarithms, the expression $\log_b(a)$ fundamentally represents:

Which of the following is the exponential form of the equation $log_{16}(4) = \frac{1}{2}$?

Convert the exponential equation $5^{-2} = \frac{1}{25}$ into its equivalent logarithmic form.

The definition of a logarithm states that $log_b(a) = c$ is equivalent to an exponential form. Which of the following expressions represents this equivalent exponential form?

Which equation represents the logarithmic form of $7^{x+1} = y$?

Find the value of $x$ in the equation $log_x(81) = 4$ by first converting it to exponential form.

Using the properties of logarithms, express $\log_7(49y^2)$ as a sum and simplify completely.

Condense the expression $\log_2(x) + \log_2(x+1) + \log_2(5)$ into a single logarithm.

Solve for $x$ in the equation $\log_6(x) + \log_6(x-5) = 2$. Remember to check for extraneous solutions.

Given that $\log_a(2) = p$ and $\log_a(3) = q$, find an expression for $\log_a(12)$ in terms of $p$ and $q$.

Which of the following statements demonstrates an incorrect application of the properties of logarithms?

Express the expression $\log_5(75) - \log_5(3)$ as a single logarithm and simplify.

Which of the following is equivalent to $\log_b\left(\frac{x^4}{y^2}\right)$?

Given that $\log_a(5) \approx 1.61$ and $\log_a(2) \approx 0.69$. Find the approximate value of $\log_a(2.5)$.

Solve for $x$ in the equation: $\log_3(x+8) - \log_3(x) = 2$.

Which expression is NOT equivalent to $\log\left(\frac{100}{x^2}\right)$?

Simplify the following logarithmic expression: $2\log_3(9x) - \log_3(x^2)$.

Given that $\log_b(5) = y$, which of the following expressions is equivalent to $\log_b(\sqrt[3]{25})$?

Solve for the variable $x$ in the equation: $3\log_2(x) = \log_2(64)$.

Using the properties of logarithms, which of the following expressions is equivalent to $\log\left(\frac{a^5}{b^2}\right)$? (Assume the base is 10).

A common mistake is to assume $\log(x^p) = (\log x)^p$. For which set of positive values of $x$ is the statement $\log_4(x^2) = (\log_4 x)^2$ actually true?