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SAT II Math II : Transformations

Study concepts, example questions & explanations for SAT II Math II

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Example Questions

Example Question #1 : Coordinate Geometry

Axes_1

Refer to the above figure.

Which of the following functions is graphed?

Possible Answers:

$\displaystyle g(x) = |x+3| + 6$

$\displaystyle g(x) = |x - 6| + 3$

$\displaystyle g(x) = |x +6| + 3$

The correct answer is not given among the other responses.

$\displaystyle g(x) = |x-3| + 6$

Correct answer:

$\displaystyle g(x) = |x - 6| + 3$

Explanation:

Below is the graph of $\displaystyle f(x) = |x|$:

Axes_1

The given graph is the graph of $\displaystyle f$ shifted 6 units left (that is, $\displaystyle -6$ unit right) and 3 units up.

The function graphed is therefore

$\displaystyle g (x) = f(x-h) + k$ where $\displaystyle h = 6, k= 3$. That is,

$\displaystyle g (x) = f(x-6) +3$

$\displaystyle g(x)= | x-6 | + 3$

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Example Question #2 : Coordinate Geometry

Axes_1

Refer to the above figure.

Which of the following functions is graphed?

Possible Answers:

$\displaystyle g(x) = |2x + 6|$

$\displaystyle g(x) = |2x| + 3$

$g(x) = \left |3x + \frac{3}{2} \right |\textup{}$ $\displaystyle g(x) = \left |3x + \frac{3}{2} \right |\textup{}$

$\displaystyle g(x) = |3x +6|$

$g(x) =\left | \frac{1}{2} x \right |+ 3$ $\displaystyle g(x) =\left | \frac{1}{2} x \right |+ 3$

Correct answer:

$\displaystyle g(x) = |2x| + 3$

Explanation:

Below is the graph of $\displaystyle f(x) = |x|$:

Axes_1

If the graph of $\displaystyle f$ is translated by shifting each point (x,y) $\displaystyle (x,y)$ to the point (2, 2y) $\displaystyle (2, 2y)$, the graph of

$\displaystyle g(x) = 2f(x)$

is formed. If the graph is then shifted upward by three units, the new graph is

$\displaystyle g(x) = 2f(x) + 3$

Since the starting graph was $\displaystyle f(x) = |x|$, the final graph is

$\displaystyle g(x) = 2 |x| + 3$, or,

$\displaystyle g(x) = |2x| + 3$

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Example Question #2 : Coordinate Geometry

Axes_1

Refer to the above figure.

Which of the following functions is graphed?

Possible Answers:

$\displaystyle g(x) = |3x+12|$

$g(x) =\left | \frac{1}{3} x+\frac{4}{3} \right |$ $\displaystyle g(x) =\left | \frac{1}{3} x+\frac{4}{3} \right |$

$g(x) =\left | \frac{1}{3} x-\frac{4}{3} \right |$ $\displaystyle g(x) =\left | \frac{1}{3} x-\frac{4}{3} \right |$

$\displaystyle g(x) =- |3x+12|$

$\displaystyle g(x) = |3x-12|$

Correct answer:

$\displaystyle g(x) = |3x-12|$

Explanation:

Below is the graph of $\displaystyle f(x) = |x|$:

Axes_1

If the graph of $\displaystyle f$ is translated by shifting each point (x,y) $\displaystyle (x,y)$ to the point (x, 3y) $\displaystyle (x, 3y)$, the graph of

$\displaystyle g(x) = 3f(x)$

is formed. If the graph is then shifted right by four units, the new graph is

$\displaystyle g(x) = 3f(x- 4)$

Since the starting graph was $\displaystyle f(x) = |x|$, the final graph is

$\displaystyle g(x) = 3 |x-4|$, or $\displaystyle g(x) = |3x-12|$

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Example Question #1 : Transformations

Axes_1

Refer to the above figure.

Which of the following functions is graphed?

Possible Answers:

$\displaystyle g(x) =-|4x |- 4$

$\displaystyle g(x) =-|4x+16|$

$g(x) =\left | \frac{1}{4}x \right |- 4$ $\displaystyle g(x) =\left | \frac{1}{4}x \right |- 4$

$g(x) =\left | \frac{1}{4}x-1 \right |$ $\displaystyle g(x) =\left | \frac{1}{4}x-1 \right |$

$g(x) =\left | \frac{1}{4}x+1 \right |$ $\displaystyle g(x) =\left | \frac{1}{4}x+1 \right |$

Correct answer:

$g(x) =\left | \frac{1}{4}x \right |- 4$ $\displaystyle g(x) =\left | \frac{1}{4}x \right |- 4$

Explanation:

Below is the graph of $\displaystyle f(x) = |x|$:

Axes_1

If the graph of $\displaystyle f$ is translated by shifting each point (x,y) $\displaystyle (x,y)$ to the point $\left ( x, \frac{1}{4}y \right )$ $\displaystyle \left ( x, \frac{1}{4}y \right )$, the graph of

$g(x) = \frac{1}{4}f(x)$ $\displaystyle g(x) = \frac{1}{4}f(x)$

is formed. If the graph is then shifted down by four units, the new graph is

$g(x) = \frac{1}{4}f(x)- 4$ $\displaystyle g(x) = \frac{1}{4}f(x)- 4$.

Since the starting graph was $\displaystyle f(x) = |x|$, the final graph is

$g(x) = \frac{1}{4}|x|- 4$ $\displaystyle g(x) = \frac{1}{4}|x|- 4$, or

$g(x) =\left | \frac{1}{4}x \right |- 4$ $\displaystyle g(x) =\left | \frac{1}{4}x \right |- 4$

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Example Question #1 : Transformations

Axes_1

Refer to the above figure.

Which of the following functions is graphed?

Possible Answers:

$g(x) =\left | \frac{1}{3} x+ \frac{2}{3} \right |$ $\displaystyle g(x) =\left | \frac{1}{3} x+ \frac{2}{3} \right |$

$\displaystyle g(x) = |3x-6|$

$g(x) =\left | \frac{1}{3} x- \frac{2}{3} \right |$ $\displaystyle g(x) =\left | \frac{1}{3} x- \frac{2}{3} \right |$

$g(x) =\left | \frac{1}{3} x \right |+2$ $\displaystyle g(x) =\left | \frac{1}{3} x \right |+2$

$\displaystyle g(x) = |3x|-2$

Correct answer:

$g(x) =\left | \frac{1}{3} x- \frac{2}{3} \right |$ $\displaystyle g(x) =\left | \frac{1}{3} x- \frac{2}{3} \right |$

Explanation:

Below is the graph of $\displaystyle f(x) = |x|$:

Axes_1

If the graph of $\displaystyle f$ is translated by shifting each point (x,y) $\displaystyle (x,y)$ to the point $\left ( x, \frac{1}{3}y \right )$ $\displaystyle \left ( x, \frac{1}{3}y \right )$, the graph of

$g(x) = \frac{1}{3}f(x)$ $\displaystyle g(x) = \frac{1}{3}f(x)$

is formed. If the graph is then shifted right by two units, the new graph is

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

Since the starting graph was $\displaystyle f(x) = |x|$, the final graph is

$g(x) = \frac{1}{3} |x-2|$ $\displaystyle g(x) = \frac{1}{3} |x-2|$, or

$g(x) =\left | \frac{1}{3} x- \frac{2}{3} \right |$ $\displaystyle g(x) =\left | \frac{1}{3} x- \frac{2}{3} \right |$

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All SAT II Math II Resources

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SAT II Math II : Transformations

Study concepts, example questions & explanations for SAT II Math II

All SAT II Math II Resources

Example Questions

Example Question #1 : Coordinate Geometry

Example Question #2 : Coordinate Geometry

Example Question #2 : Coordinate Geometry

Example Question #1 : Transformations

Example Question #1 : Transformations

All SAT II Math II Resources

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