Multiple Choice

1. The quadratic function $f(x) = x^2$ is first reflected across the x-axis, then stretched vertically by a factor of $3$, then translated $2$ units to the right and $5$ units up. What is the equation of the new function, $g(x)$?

   $g(x) = 3(-x-2)^2 + 5$

   $g(x) = -3(x+2)^2 + 5$

   $g(x) = -3(x-2)^2 + 5$

   $g(x) = 3(x-2)^2 - 5$

2. The function $f(x) = \sqrt{x}$ undergoes the following transformations in order: a horizontal stretch by a factor of $2$, a reflection across the y-axis, a translation $1$ unit to the left, and a translation $3$ units down. What is the equation of the transformed function, $g(x)$?

   $g(x) = \sqrt{-2(x+1)} - 3$

   $g(x) = \sqrt{-\frac{1}{2}(x+1)} - 3$

   $g(x) = \sqrt{-2x+1} - 3$

   $g(x) = \sqrt{-\frac{1}{2}(x-1)} - 3$

3. A quadratic function $f(x) = x^2$ has its vertex at $(0,0)$. After a series of transformations, the new function $g(x)$ has its vertex at $(-3, 4)$ and passes through the point $(-2, 2)$. What is the equation of $g(x)$?

   $g(x) = -2(x+3)^2 + 4$

   $g(x) = -2(x-3)^2 + 4$

   $g(x) = -2(x+3)^2 - 4$

   $g(x) = -(x-3)^2 + 4$

4. The function $f(x) = \sqrt{x}$ is transformed such that its new starting point is $(5, -2)$ and it passes through the point $(6, -3)$. If there is no horizontal reflection or stretch/compression, what is the equation of the transformed function $g(x)$?

   $g(x) = -\sqrt{x+5} - 2$

   $g(x) = -2\sqrt{x-5} - 2$

   $g(x) = -\sqrt{x-5} - 2$

   $g(x) = \sqrt{-(x-5)} - 2$

5. A parabola $y = ax^2 + bx + c$ has its vertex at $(0,0)$. After being transformed to $y = -2(x+4)^2 + 1$, describe the sequence of transformations.

   Vertical stretch by $2$, reflect across x-axis, translate $4$ units right, translate $1$ unit up.

   Reflect across x-axis, vertical stretch by $2$, translate $4$ units right, translate $1$ unit up.

   Reflect across x-axis, vertical stretch by $2$, translate $4$ units left, translate $1$ unit down.

   Reflect across x-axis, vertical stretch by $2$, translate $4$ units left, translate $1$ unit up.

6. The function $f(x) = \sqrt{x}$ is transformed by a horizontal compression by a factor of $1/3$, a reflection across the y-axis, and then a translation $2$ units to the right. What is the equation of $g(x)$?

   $g(x) = \sqrt{-3x+2}$

   $g(x) = \sqrt{-3(x+2)}$

   $g(x) = \sqrt{-3(x-2)}$

   $g(x) = \sqrt{-\frac{1}{3}(x-2)}$

7. The parabola $f(x) = x^2$ has points $(1,1)$ and $(2,4)$. If $f(x)$ is transformed to $g(x) = - (2(x-1))^2 + 3$, what are the new coordinates of the points $(1,1)$ and $(2,4)$ respectively?

   $(0.5, 4)$ and $(1.5, -13)$

   $(1.5, 4)$ and $(2, -1)$

   $(1.5, 2)$ and $(2, -1)$

   $(0.5, 2)$ and $(1, -13)$

8. A function $f(x)$ is transformed to $g(x) = 2\sqrt{-(x+3)} - 1$. If the transformations applied were a reflection across the y-axis, a vertical stretch by a factor of $2$, a translation $3$ units left, and a translation $1$ unit down, what was the original function $f(x)$?

   $f(x) = \sqrt{x}$

   $f(x) = \sqrt{-x}$

   $f(x) = \sqrt{x-3}$

   $f(x) = 2\sqrt{x}$

9. The function $f(x) = x^2$ is transformed by a horizontal compression by a factor of $1/2$, then translated $3$ units to the right, then reflected across the x-axis, then translated $4$ units up. What is the equation of the new function, $g(x)$?

   $g(x) = -(2(x+3))^2 + 4$

   $g(x) = -(2x-3)^2 + 4$

   $g(x) = -(2(x-3))^2 + 4$

   $g(x) = -(2x+6)^2 + 4$

10. The function $g(x) = -3\sqrt{\frac{1}{4}(x+2)} + 5$ is a transformation of $f(x) = \sqrt{x}$. Describe the transformations applied.

    Reflection across x-axis, vertical stretch by $3$, horizontal stretch by $4$, translate $2$ units left, translate $5$ units up.

    Reflection across x-axis, vertical stretch by $3$, horizontal compression by $4$, translate $2$ units left, translate $5$ units up.

    Vertical stretch by $3$, reflection across y-axis, horizontal stretch by $4$, translate $2$ units left, translate $5$ units up.

    Reflection across x-axis, vertical stretch by $3$, horizontal stretch by $4$, translate $2$ units right, translate $5$ units up.

11. The quadratic function $f(x) = x^2 - 4$ has $x$-intercepts at $(-2,0)$ and $(2,0)$. If $f(x)$ is transformed by a vertical stretch by a factor of $2$, a horizontal translation $1$ unit left, and a vertical translation $3$ units up, what is the new equation $g(x)$?

    $g(x) = 2(x+1)^2 - 1$

    $g(x) = 2(x+1)^2 - 5$

    $g(x) = 2(x+1)^2 - 4$

    $g(x) = 2(x-1)^2 - 1$

12. The function $f(x) = \sqrt{x}$ is transformed into $g(x)$ by a reflection across the x-axis, a vertical stretch by a factor of $2$, a horizontal stretch by a factor of $4$, and then translated $3$ units to the right and $1$ unit up. What is the equation of $g(x)$?

    $g(x) = -2\sqrt{4x-3} + 1$

    $g(x) = -2\sqrt{\frac{1}{4}(x-3)} + 1$

    $g(x) = -2\sqrt{4(x-3)} + 1$

    $g(x) = 2\sqrt{\frac{1}{4}(x-3)} + 1$

13. Given the quadratic function $f(x) = (x-1)^2 + 3$. It is horizontally stretched by a factor of $0.5$, reflected across the y-axis, then translated $2$ units left and $4$ units down. What is the equation of the transformed function, $g(x)$?

    $g(x) = (-2(x+2)+1)^2 - 1$

    $g(x) = (-2(x+2)-1)^2 - 1$

    $g(x) = (2(x-2)+1)^2 - 1$

    $g(x) = (-2x+4-1)^2 - 1$

14. The function $y = \sqrt{x}$ is transformed to $y = -2\sqrt{-(3x+6)} + 1$. Describe the sequence of transformations.

    Reflection across x-axis, vertical stretch by $2$, horizontal compression by $3$, reflection across y-axis, translate $6$ units left, translate $1$ unit up.

    Reflection across x-axis, vertical stretch by $2$, horizontal compression by $3$, reflection across y-axis, translate $2$ units left, translate $1$ unit up.

    Reflection across y-axis, horizontal compression by $3$, reflection across x-axis, vertical stretch by $2$, translate $2$ units left, translate $1$ unit up.

    Vertical stretch by $2$, reflection across y-axis, horizontal compression by $3$, reflection across y-axis, translate $6$ units left, translate $1$ unit up.

15. A quadratic function $f(x) = x^2$ is transformed. Its vertex is shifted to $(4, -7)$. It is also reflected across the x-axis and vertically stretched by a factor of $0.5$. What is the new equation of the function $g(x)$?

    $g(x) = -2(x+4)^2 - 7$

    $g(x) = -0.5(x+4)^2 - 7$

    $g(x) = -0.5(x-4)^2 - 7$

    $g(x) = 0.5(x-4)^2 - 7$

16. The function $f(x) = \sqrt{x}$ is transformed by translating $5$ units right, then horizontally stretching by a factor of $2$, then vertically compressing by a factor of $1/3$, then translating $1$ unit up. What is the equation of the new function $g(x)$?

    $g(x) = \frac{1}{3}\sqrt{\frac{1}{2}(x-5)} + 1$

    $g(x) = \frac{1}{3}\sqrt{\frac{1}{2}x-5} + 1$

    $g(x) = \frac{1}{3}\sqrt{2(x-5)} + 1$

    $g(x) = \frac{1}{3}\sqrt{2x-5} + 1$

17. A quadratic function $f(x) = x^2$ is stretched vertically by a factor of $2$ and horizontally by a factor of $3$. What is the equation of the new function, $g(x)$?

    $g(x) = 6x^2$

    $g(x) = 2(\frac{x}{3x})^2$

    $g(x) = 2(\frac{x}{3})^2$

    $g(x) = 2(3x)^2$

18. A root function $f(x)$ is transformed by a reflection across the x-axis, a vertical stretch by $3$, a horizontal compression by $2$, a translation $4$ units left, and $2$ units down. The resulting function is $g(x) = -3\sqrt{2(x+4)} - 2$. What was the original function $f(x)$?

    $f(x) = \sqrt{x}$

    $f(x) = \sqrt{x+4}$

    $f(x) = \sqrt{2x}$

    $f(x) = \sqrt{x-4}$

19. The quadratic function $f(x) = x^2$ is first reflected across the y-axis, then across the x-axis, then translated $3$ units up and $1$ unit left. What is the equation of the new function, $g(x)$?

    $g(x) = (x+1)^2+3$

    $g(x) = -(x-1)^2+3$

    $g(x) = -(-x+1)^2+3$

    $g(x) = -(x+1)^2+3$

20. The graph of $y = \sqrt{x}$ passes through the point $(9,3)$. After a series of transformations, the new function $g(x)$ has its starting point at $(-1, 4)$ and passes through the point $(0, 1)$. What is the equation of $g(x)$?

    $g(x) = -3\sqrt{x-1} + 4$

    $g(x) = 3\sqrt{-(x+1)} + 4$

    $g(x) = -\sqrt{x+1} + 4$

    $g(x) = -3\sqrt{x+1} + 4$