Q: A student graphs $f(x) = \sqrt{x}$ and $g(x) = \sqrt{x} + k$. If $k < 0$, what happens to the graph of $f(x)$?

It is shifted downward by $|k|$ units.

Question 1

Which of the following describes the effect of multiplying the function $f(x) = \sqrt{x}$ by a constant $a > 1$?

Accepted Answer

It stretches the graph vertically by a factor of $a$.

Question 2

Given $f(x) = |x|$, what is the effect of the transformation $g(x) = |x| + 2$?

Accepted Answer

It shifts the graph of $f(x)$ upward by 2 units.

Question 3

If $h(x) = \sqrt{x}$ and $k(x) = \frac{1}{2}\sqrt{x}$, which statement best describes the transformation from $h(x)$ to $k(x)$?

Accepted Answer

The graph of $h(x)$ is shrunk vertically by a factor of $\frac{1}{2}$.

Question 4

Suppose $f(x) = \sqrt{x}$ and $g(x) = \sqrt{x} + 3$. What is the $y$-intercept of $g(x)$?

Accepted Answer

$3$

Question 5

Which transformation will shift the graph of $f(x) = |x|$ downward by 2 units?

Accepted Answer

$g(x) = |x| - 2$

Question 6

If $f(x) = \sqrt{x}$, which of the following represents a vertical stretch by a factor of $3$ and a shift upward by $2$ units?

Accepted Answer

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

Question 7

A medical researcher models the recovery rate of a drug with $f(x) = \sqrt{x}$, where $x$ is the number of days since treatment. If the recovery rate is observed to be half as fast, which function best models the new recovery rate?

Accepted Answer

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

Question 8

Which of the following transformations will reflect the graph of $f(x) = \sqrt{x}$ across the $x$-axis?

Accepted Answer

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

Question 9

If $f(x) = |x|$, which transformation results in a graph that is both vertically stretched by a factor of $2$ and shifted downward by $3$ units?

Accepted Answer

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

Question 10

A company models its profit with $f(x) = |x|$, where $x$ is the number of products sold. If the company increases its base profit by $5$ units, which function models the new profit?

Accepted Answer

$g(x) = |x| + 5$

Question 11

Given $f(x) = \sqrt{x}$, which transformation will shrink the graph vertically by a factor of $\frac{1}{4}$ and shift it upward by $1$ unit?

Accepted Answer

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

Question 12

If $f(x) = |x|$, what is the effect of the transformation $g(x) = |x| - 2$?

Accepted Answer

It shifts the graph of $f(x)$ downward by 2 units.

Question 13

Which of the following best describes the transformation from $f(x) = \sqrt{x}$ to $g(x) = \sqrt{x} - 2$?

Accepted Answer

The graph is shifted downward by 2 units.

Question 14

A scientist models the growth of bacteria with $f(x) = \sqrt{x}$, where $x$ is time in hours. If the bacteria grow twice as fast, which function best models the new growth?

Accepted Answer

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

Question 15

If $f(x) = |x|$, which transformation will reflect the graph across the $x$-axis?

Accepted Answer

$g(x) = -|x|$

Question 16

Which of the following transformations will shift the graph of $f(x) = \sqrt{x}$ upward by $4$ units?

Accepted Answer

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

Question 17

If $f(x) = |x|$, which transformation will shrink the graph vertically by a factor of $\frac{1}{3}$?

Accepted Answer

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

Question 18

A student graphs $f(x) = \sqrt{x}$ and $g(x) = \sqrt{x} + k$. If $k < 0$, what happens to the graph of $f(x)$?

Accepted Answer

It is shifted downward by $|k|$ units.

Question 19

Which transformation will both reflect the graph of $f(x) = \sqrt{x}$ across the $x$-axis and shift it upward by $3$ units?

Accepted Answer

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

Transformations of Functions: Stretching, Shrinking, and Vertical Translation

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