College Algebra Quiz: Graphing From Parent Functions

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Graphing From Parent FunctionsQuestion 1 of 8

If $f(x) = x^3$ and $g(x) = \frac{1}{2}(x + 1)^3 - 4$ , then $g(x)$ can be described as $f(x)$ after which transformations?

Horizontal compression by factor

\frac{1}{2}

, shift left 1 unit, shift down 4 units

Vertical compression by factor

\frac{1}{2}

, shift left 1 unit, shift down 4 units

Vertical compression by factor

\frac{1}{2}

, shift right 1 unit, shift down 4 units

Horizontal compression by factor

\frac{1}{2}

, shift right 1 unit, shift up 4 units

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College Algebra Quiz

College Algebra Quiz: Graphing From Parent Functions

Practice Graphing From Parent Functions in College Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Graphing From Parent Functions, giving you a quick way to practice the rules, question types, and explanations that matter most for College Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

If $f(x) = x^3$ and $g(x) = \frac{1}{2}(x + 1)^3 - 4$ , then $g(x)$ can be described as $f(x)$ after which transformations?

Horizontal compression by factor $\frac{1}{2}$ , shift left 1 unit, shift down 4 units
Vertical compression by factor $\frac{1}{2}$ , shift left 1 unit, shift down 4 units (correct answer)
Vertical compression by factor $\frac{1}{2}$ , shift right 1 unit, shift down 4 units
Horizontal compression by factor $\frac{1}{2}$ , shift right 1 unit, shift up 4 units

Explanation: Starting with

f(x) = x^3

: The term

(x + 1)^3

represents a horizontal shift left 1 unit. The coefficient

\frac{1}{2}

represents a vertical compression by factor

\frac{1}{2}

. The

-4

represents a vertical shift down 4 units. So

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

. Choice A confuses vertical and horizontal compression. Choice C shifts right instead of left. Choice D has multiple errors: wrong compression type, wrong horizontal direction, and wrong vertical direction.

Question 2

The parent function $f(x) = \frac{1}{x}$ is transformed to create $h(x) = \frac{-2}{x+3} + 1$ . What are the equations of the vertical and horizontal asymptotes of $h(x)$ ?

Vertical: $x = 3$ , Horizontal: $y = 1$
Vertical: $x = -3$ , Horizontal: $y = -1$
Vertical: $x = 3$ , Horizontal: $y = -1$
Vertical: $x = -3$ , Horizontal: $y = 1$ (correct answer)

Explanation: When working with rational function transformations, you need to identify how the parent function

f(x) = \frac{1}{x}

has been shifted to find the new asymptotes. The transformed function

h(x) = \frac{-2}{x+3} + 1

shows two key transformations from the parent function. The vertical asymptote occurs where the denominator equals zero, so you set

x + 3 = 0

, which gives

x = -3

. The horizontal asymptote is determined by the constant term added outside the fraction, which is

+1

, so the horizontal asymptote is

y = 1

. Looking at the incorrect choices: Choice A places the vertical asymptote at

x = 3

, which incorrectly assumes the shift is in the positive direction rather than recognizing that

x + 3 = 0

means

x = -3

. Choice B correctly identifies the vertical asymptote as

x = -3

but incorrectly places the horizontal asymptote at

y = -1

, confusing the sign of the vertical shift. Choice C combines both errors—wrong vertical asymptote and wrong horizontal asymptote. The correct answer is D: the vertical asymptote is

x = -3

and the horizontal asymptote is

y = 1

. Study tip: For rational function transformations in the form

\frac{a}{x-h} + k

, remember that the vertical asymptote is

x = h

(set the denominator to zero) and the horizontal asymptote is

y = k

(the vertical shift). Watch the signs carefully—

x + 3

means

h = -3

, not

+3

Question 3

Consider the function $m(x) = -\frac{1}{2}(x+2)^3 + 1$ . If the point $(a, b)$ lies on the graph of the parent function $f(x) = x^3$ , what point lies on the graph of $m(x)$ ?

$(-\frac{1}{2}(a + 2), b + 1)$
$(a + 2, -\frac{1}{2}b + 1)$
$(a - 2, \frac{1}{2}b + 1)$
$(a - 2, -\frac{1}{2}b + 1)$ (correct answer)

Explanation: When you see a function transformation problem like this, you need to track how each transformation affects the coordinates of points on the original function. Let's analyze

m(x) = -\frac{1}{2}(x+2)^3 + 1

compared to the parent function

f(x) = x^3

. This involves three transformations applied in sequence:

The $(x+2)$ shifts the graph 2 units left
The $-\frac{1}{2}$ coefficient reflects over the x-axis and compresses vertically by factor $\frac{1}{2}$
The $+1$ shifts the graph 1 unit up

If point

(a,b)

is on

f(x) = x^3

, then

b = a^3

. To find the corresponding point on

m(x)

, apply the transformations to the coordinates:

Horizontal shift left 2: $x$ -coordinate becomes $a - 2$
Vertical reflection and compression: $y$ -coordinate becomes $-\frac{1}{2}b$
Vertical shift up 1: $y$ -coordinate becomes $-\frac{1}{2}b + 1$

The transformed point is

(a-2, -\frac{1}{2}b + 1)

, which matches choice D. Choice A incorrectly applies the transformations to get the coordinates rather than finding where the point moves. Choice B has the correct

y

-coordinate transformation but wrong

x

-coordinate (

a+2

instead of

a-2

). Choice C misses the reflection, keeping

+\frac{1}{2}b

instead of

-\frac{1}{2}b

. Strategy tip: For transformation problems, systematically apply each change to the coordinates in order. Remember that

f(x+h)

shifts left by

h

units, which often trips students up since the sign seems backwards.

Question 4

The function $p(x) = 3^{-x+2} + 1$ is a transformation of the parent exponential function $f(x) = 3^x$ . Which point on the graph of $f(x)$ corresponds to the point $(3, 2)$ on the graph of $p(x)$ ?

$(1, 1)$
$(5, 1)$
$(-1, 1)$ (correct answer)
$(1, 3)$

Explanation: We can rewrite

p(x) = 3^{-x+2} + 1 = 3^{2-x} + 1

. If

(3, 2)

is on

p(x)

, we verify:

p(3) = 3^{2-3} + 1 = 3^{-1} + 1 = \frac{1}{3} + 1 = \frac{4}{3}

. This doesn't equal 2, so let me recalculate. Actually,

p(3) = 3^{-3+2} + 1 = 3^{-1} + 1 = \frac{1}{3} + 1 = \frac{4}{3} \neq 2

. Let me check which x gives

p(x) = 2

3^{-x+2} + 1 = 2

, so

3^{-x+2} = 1 = 3^0

, thus

-x+2 = 0

and

x = 2

. So

(2, 2)

is on

p(x)

. Working backwards from this point: the corresponding point on

f(x) = 3^x

is found by reversing transformations. Since

p(x) = 3^{-(x-2)} + 1

, this involves reflection across y-axis and shift up 1. Reversing:

(2,2) \rightarrow (2,1) \rightarrow (-2,1)

f(x)

. But this doesn't match our given point. Let me verify the original constraint matches an answer choice by checking

f(-1) = 3^{-1} = \frac{1}{3}

, and working forward through transformations.

Question 5

Consider the function $h(x) = \sqrt{4 - 2x} + 3$ . This function can be obtained by transforming the parent function $f(x) = \sqrt{x}$ . What is the domain of $h(x)$ ?

$(-\infty, 2]$ (correct answer)
$[2, \infty)$
$[-2, \infty)$
$(-\infty, -2]$

Explanation: For

h(x) = \sqrt{4 - 2x} + 3

to be defined, we need

4 - 2x \geq 0

. Solving:

4 - 2x \geq 0 \Rightarrow 4 \geq 2x \Rightarrow 2 \geq x

, so

x \leq 2

. The domain is

(-\infty, 2]

. Choice B reverses the inequality. Choice C uses

x \geq -2

, which comes from incorrectly solving

4 - 2x \geq 0

-2x \geq -4

without flipping the inequality sign. Choice D combines both errors.

Question 6

Which of the following functions represents a horizontal stretch of $f(x) = (x-1)^2 + 2$ by a factor of 3?

$g(x) = 3(x-1)^2 + 2$
$g(x) = (3x-1)^2 + 2$
$g(x) = \left(\frac{x}{3}-1\right)^2 + 2$ (correct answer)
$g(x) = \left(\frac{x-1}{3}\right)^2 + 2$

Explanation: A horizontal stretch by factor 3 means replacing

x

with

\frac{x}{3}

in the function. Starting with

f(x) = (x-1)^2 + 2

, we get

g(x) = \left(\frac{x}{3}-1\right)^2 + 2

. Choice A is a vertical stretch by factor 3. Choice B represents a horizontal compression by factor

\frac{1}{3}

. Choice D incorrectly applies the stretch factor to the entire expression

(x-1)

rather than just

x

, which would not preserve the horizontal shift properly.

Question 7

The function $g(x) = -2|x + 3| - 1$ is a transformation of the parent function $f(x) = |x|$ . Which sequence of transformations, applied in order, produces $g(x)$ from $f(x)$ ?

Shift left 3 units, reflect across x-axis, vertical stretch by factor 2, shift down 1 unit (correct answer)
Shift right 3 units, vertical stretch by factor 2, reflect across x-axis, shift down 1 unit
Reflect across x-axis, shift left 3 units, vertical stretch by factor 2, shift down 1 unit
Vertical stretch by factor 2, shift left 3 units, reflect across x-axis, shift up 1 unit

Explanation: To transform

f(x) = |x|

into

g(x) = -2|x + 3| - 1

: First, shift left 3 units to get

|x + 3|

. Then reflect across the x-axis and apply vertical stretch by factor 2 to get

-2|x + 3|

. Finally, shift down 1 unit to get

-2|x + 3| - 1

. Choice B incorrectly shifts right instead of left. Choice C applies the reflection before the horizontal shift, which would give a different result. Choice D shifts up instead of down.

Question 8

If $g(x) = 2\sqrt{x-4} - 3$ and $f(x) = \sqrt{x}$ , which statement about the range of $g(x)$ compared to the range of $f(x)$ is correct?

The range of $g(x)$ is $[0, \infty)$ , the same as $f(x)$
The range of $g(x)$ is $[-3, \infty)$ , shifted down 3 units from $f(x)$ (correct answer)
The range of $g(x)$ is $[4, \infty)$ , shifted right 4 units from $f(x)$
The range of $g(x)$ is $[1, \infty)$ , reflecting the combined transformations

Explanation: The parent function

f(x) = \sqrt{x}

has range

[0, \infty)

. For

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

: The horizontal shift (

x-4

) affects the domain but not the range. The vertical stretch by factor 2 would change

[0, \infty)

[0, \infty)

. The vertical shift down 3 units changes the range to

[-3, \infty)

. Choice A ignores the vertical shift. Choice C confuses domain and range. Choice D incorrectly calculates the effect of the transformations on the range.