Multiple Choice

1. Find the equation of the image of the graph $y = x^2$ when the following sequence of transformations has been applied: a translation of $3$ units in the positive direction of the $y$-axis followed by a dilation of factor $2$ from the $x$-axis.

   $y = 2x^2 + 6$

   $y = 2(x-3)^2$

   $y = 2x^2 + 3$

   $y = (2x+3)^2$

2. Determine the equation of the graph formed by applying the following transformations to $y = \sqrt{x}$: a reflection in the $x$-axis followed by a translation of $4$ units in the positive direction of the $x$-axis.

   $y = \sqrt{-x-4}$

   $y = -\sqrt{x+4}$

   $y = -\sqrt{x-4}$

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

3. What is the equation of the image of $y = |x|$ after a dilation of factor $3$ from the $y$-axis followed by a translation of $2$ units in the negative direction of the $y$-axis?

   $y = \tfrac{1}{3}|x-2|$

   $y = 3|x| - 2$

   $y = |3x| - 2$

   $y = \left|\tfrac{1}{3}x\right| - 2$

4. The graph of $y = \frac{1}{x}$ undergoes a translation of $1$ unit in the negative direction of the $x$-axis, then a reflection in the $y$-axis. Find the resulting equation.

   $y = \frac{1}{x-1}$

   $y = \frac{1}{-(x+1)}$

   $y = \frac{1}{x+1}$

   $y = \frac{1}{1-x}$

5. The graph of $y = x^3$ is reflected in the $x$-axis, then translated $5$ units in the negative direction of the $y$-axis. What is the new equation?

   $y = (x+5)^3$

   $y = -x^3 - 5$

   $y = (-x)^3 - 5$

   $y = -(x-5)^3$

6. Find the equation of the image of $y = 2^x$ after a dilation of factor $\frac{1}{2}$ from the $y$-axis followed by a translation of $1$ unit in the positive direction of the $x$-axis.

   $y = 2^{\frac{1}{2}(x-1)}$

   $y = 2^{\frac{1}{2}x+1}$

   $y = 2^{2(x-1)}$

   $y = 2^{2x+1}$

7. The graph of $y = x^2$ is translated $2$ units in the negative direction of the $x$-axis, then reflected in the $y$-axis. What is the equation of the transformed graph?

   $y = -(x-2)^2$

   $y = (-x-2)^2$

   $y = (-x+2)^2$

   $y = (x+2)^2$

8. The graph of $y = \sqrt{x}$ undergoes a reflection in the $x$-axis, then a dilation of factor $3$ from the $x$-axis. Find the resulting equation.

   $y = -3\sqrt{-x}$

   $y = -3\sqrt{x}$

   $y = \tfrac{1}{3}\sqrt{x}$

   $y = \sqrt{-3x}$

9. What is the equation of the image of $y = |x|$ after a dilation of factor $2$ from the $y$-axis followed by a translation of $5$ units in the positive direction of the $y$-axis?

   $y = \left|\tfrac{1}{2}x\right| + 5$

   $y = 2|x| + 5$

   $y = |2x| + 5$

   $y = |x+2| + 5$

10. The graph of $y = \frac{1}{x}$ is dilated by a factor of $\frac{1}{2}$ from the $x$-axis, then translated $3$ units in the positive direction of the $x$-axis. Find the resulting equation.

    $y = \frac{2}{x-3}$

    $y = \frac{1}{2(x+3)}$

    $y = \frac{1}{2x} + 3$

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

11. The graph of $y = x^3$ is translated $1$ unit in the negative direction of the $y$-axis, then reflected in the $x$-axis. What is the new equation?

    $y = -x^3 - 1$

    $y = -(x-1)^3$

    $y = (-x-1)^3$

    $y = -x^3 + 1$

12. The graph of $y = 2^x$ is reflected in the $y$-axis, then reflected in the $x$-axis. Find the resulting equation.

    $y = -2^{-x}$

    $y = 2^{-x}$

    $y = -2^x + 1$

    $y = -2^x$

13. Find the equation of the image of the graph $y = x^2$ after a translation of $2$ units in the positive direction of the $x$-axis and $3$ units in the negative direction of the $y$-axis, followed by a dilation of factor $2$ from the $x$-axis.

    $y = 2(x-2)^2 - 3$

    $y = 2(x+2)^2 - 6$

    $y = 2(x-2)^2 - 9$

    $y = 2(x-2)^2 - 6$

14. The graph of $y = \sqrt{x}$ undergoes a dilation of factor $\frac{1}{3}$ from the $y$-axis, then a translation of $1$ unit in the negative direction of the $x$-axis and $2$ units in the positive direction of the $y$-axis.

    $y = \sqrt{3x+1} + 2$

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

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

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

15. (\emph{Revised options to resolve ambiguity}) What is the equation of the image of $y = |x|$ after a reflection in the $y$-axis, followed by a dilation of factor $2$ from the $x$-axis, then a translation of $4$ units in the negative direction of the $y$-axis?

    $y = 2|x| - 4$

    $y = 2|-x| + 4$

    $y = 2|x+4|$

    $y = -2|x| - 4$

16. The graph of $y = \frac{1}{x}$ is dilated by a factor of $3$ from the $x$-axis, reflected in the $x$-axis, then translated $2$ units in the positive direction of the $x$-axis. Find the resulting equation.

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

    $y = \frac{-3}{x-2}$

    $y = -\frac{3}{x+2}$

    $y = \frac{-3}{x} + 2$

17. The graph of $y = x^3$ undergoes a dilation of factor $2$ from the $y$-axis, then a translation of $3$ units in the negative direction of the $x$-axis and $1$ unit in the positive direction of the $y$-axis.

    $y = (\tfrac{1}{2}x+3)^3 + 1$

    $y = 2(\tfrac{1}{2}(x+3))^3 + 1$

    $y = (\tfrac{1}{2}(x+3))^3 + 1$

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

18. The graph of $y = 2^x$ is reflected in the $x$-axis, translated $2$ units in the positive direction of the $y$-axis, then dilated by a factor of $\frac{1}{2}$ from the $y$-axis.

    $y = -2^{2(x+2)}$

    $y = -2^{2x} + 2$

    $y = -2^{x/2} + 2$

    $y = -2^{-2x} + 2$

19. Find the equation of the image of the graph $y = x^2$ after a dilation of factor $\frac{1}{2}$ from the $x$-axis, followed by a dilation of factor $3$ from the $y$-axis, then a translation of $1$ unit in the negative direction of the $x$-axis.

    $y = \frac{1}{2}\left(\frac{1}{3}(x+1)\right)^2$

    $y = \frac{1}{2}\left(\frac{1}{3}x-1\right)^2$

    $y = \frac{1}{2}\big(3(x+1)\big)^2$

    $y = \frac{1}{2}\left(\frac{1}{3}(x-1)\right)^2$

20. The graph of $y = \sqrt{x}$ is reflected in the $x$-axis, reflected in the $y$-axis, then translated $4$ units in the positive direction of the $x$-axis and $2$ units in the negative direction of the $y$-axis.

    $y = \sqrt{-(x-4)}-2$

    $y = -\sqrt{-x-4}-2$

    $y = -\sqrt{-(x-4)}-2$

    $y = -\sqrt{-(x+4)}-2$