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Quizzes > Mathematics

Parent Functions Test: Graph Transformations

Quick, free parent function practice. Instant results.

Editorial: Review CompletedCreated By: Chirag AgrawalUpdated Aug 26, 2025
Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art illustration showing linear and cubic graphs on teal background for parent functions quiz

This quiz helps you identify parent functions and understand basic graph transformations, from linear to cubic. Answer each question to match equations with their graphs and spot shifts, stretches, and reflections. If you want more support, try our graphing functions practice, review the vertical line test, or brush up with function notation practice.

What is the parent function of a linear function?
f(x)=x^2
f(x)=x
f(x)=?x
f(x)=x^3
The simplest form of any linear function is f(x)=x, which is a straight line with slope 1 through the origin. All other linear functions can be obtained by scaling and translating this parent function. Its graph is the 45-degree line in the coordinate plane. Learn more about linear functions.
What is the domain of the parent function f(x)=x?
Only integers
x ? 0
All real numbers
x ? 0
The parent function f(x)=x is defined for every real input, so its domain is all real numbers. There are no restrictions like square roots or denominators that would limit x. This is typical of simple polynomial functions of first degree. .
What is the parent function of a cubic function?
f(x)=x^2
f(x)=x^3
f(x)=1/x
f(x)=?x
The simplest cubic function is f(x)=x^3, which serves as the parent cubic. It has an S-shaped curve passing through the origin. All other cubic functions are derived by scaling or translating this basic form. Learn more about cubic functions.
What is the range of the function f(x)=x?
Only positive values
y ? 0
All real numbers
y ? 0
Since f(x)=x produces every real number as x varies over all real inputs, its range is all real numbers. There are no restrictions that prevent the output from taking any real value. This holds for any linear function with nonzero slope. .
Which of the following best describes the graph of the parent function f(x)=x^3?
A V-shaped graph
A straight line at 45°
An S-shaped curve through the origin
A U-shaped parabola
The parent cubic function f(x)=x^3 has an S-shaped curve that passes through the origin, rising to the right and falling to the left. This shape is characteristic of odd-degree polynomials with positive leading coefficient. Parabolas and V-shapes belong to different parent functions. Cubic function graphs.
Which parent function always yields non-negative outputs for all x?
f(x)=1/x
f(x)=|x|
f(x)=x
f(x)=x^3
The absolute value parent function f(x)=|x| outputs the distance from zero, which is always non-negative. It creates a V-shaped graph with its vertex at the origin. Linear and cubic parents can output negative values, and a reciprocal can also be negative. Absolute value function.
What is the domain of the parent function f(x)=x^3?
All real numbers
x ? 0
No real solutions
x ? 0
A cubic polynomial like f(x)=x^3 is defined for every real x, so its domain is all real numbers. There are no square roots or denominators that would limit input values. This is a property of all polynomial functions. .
Which function is the parent of a quadratic function?
f(x)=x^2
f(x)=x^3
f(x)=1/x
f(x)=x
The function f(x)=x^2 is the simplest form of a quadratic function, forming the basic U-shaped parabola. Other quadratic functions arise by translating and scaling this parent. Linear and cubic parents follow different polynomial degrees. Quadratic functions explained.
What transformation occurs when the parent function y=x is changed to y=3x?
Vertical compression by a factor of 3
Vertical stretch by a factor of 3
Translation up by 3 units
Horizontal stretch by a factor of 3
Multiplying the output by 3 stretches the graph of y=x vertically by a factor of 3, making it steeper. Horizontal stretches involve multiplying the input inside the function. This is a standard vertical dilation. .
What transformation is represented by g(x)=(x-2)^3?
Shift up by 2 units
Shift down by 2 units
Shift left by 2 units
Shift right by 2 units
Replacing x with (x-2) shifts the graph of the parent cubic to the right by 2 units. Horizontal translations subtract inside the function. The shape and orientation remain unchanged. .
Which sequence of transformations maps y=x^3 to y=(x+4)^3 - 1?
Shift right 4 units, then down 1 unit
Shift left 4 units, then up 1 unit
Shift left 4 units, then down 1 unit
Shift right 4 units, then up 1 unit
Adding 4 inside the cube shifts the graph left by 4 units. Subtracting 1 on the outside moves it down 1 unit. The core shape of the cubic remains intact. Transforming function graphs.
What is the y-intercept of the function f(x)=x^3 - 5?
(0, 5)
(-5, 0)
(0, -5)
(5, 0)
The y-intercept occurs when x=0, so f(0)=0^3 - 5 = -5. Thus the point is (0, -5). This is found by substituting x=0 into any function. .
Which transformation occurs when f(x)=-(x-1)^3?
Reflect across the y-axis and shift left 1 unit
Reflect across the x-axis and shift left 1 unit
Reflect across the x-axis and shift right 1 unit
Reflect across the y-axis and shift right 1 unit
The negative sign outside reflects the graph across the x-axis. Replacing x by (x-1) then shifts it right by 1 unit. Reflections and translations combine in this way. .
What is the domain of the function f(x)=?x?
All real numbers
x > 0
x ? 0
x ? 0
The square root function ?x is only defined for non-negative x values, so its domain is x ? 0. Negative inputs produce imaginary outputs, which are not in the real-valued parent function. .
What is the range of the function f(x)=?x - 3?
All real numbers
y ? -3
y ? -3
y > -3
Since ?x ? 0 for x ? 0, subtracting 3 shifts the entire range down by 3. Thus the smallest output is -3, and the range is y ? -3. No values less than -3 occur. .
Which of the following is the equation of a cubic function shifted up by 3 units?
(x + 3)^3
x^3 + 3
(x - 3)^3
x^3 - 3
Adding 3 on the outside of x^3 shifts the graph vertically upward by 3 units. Inside shifts affect the x-axis, not the y-axis. Thus f(x)=x^3+3 is the correct transformed cubic. .
Identify the values of a, h, and k in the function f(x)=2(x+3)^3 - 4.
a=2, h=-3, k=-4
a=-2, h=-3, k=4
a=2, h=3, k=4
a=-2, h=3, k=-4
In the form a(x-h)^3 + k, a=2, h=-3 (since x+3 gives h=-3), and k=-4. These parameters scale, shift, and translate the parent cubic. Recognizing the form helps identify each transformation. .
Solve the equation x^3 - 8 = 0 for x.
-2
0
2
4
Setting x^3 - 8 = 0 gives x^3 = 8, so x = ?8 = 2. This is the real root of the cubic. Other roots would be complex, but the parent function focuses on the real solution. .
What is the inverse function of f(x)=x^3?
f?¹(x)=x^2
f?¹(x)=-?x
f?¹(x)=?x
f?¹(x)=?x
The inverse of f(x)=x^3 swaps x and y and solves for y: x = y^3 yields y = ?x. Thus f?¹(x)=?x. Other options do not reverse the cube function. Inverse functions explained.
What is (f?g)(x) if f(x)=x^3 and g(x)=x+2?
(x)^3 + 2
x^3 + 2
x^3 + 6
(x+2)^3
The composition f(g(x)) means you cube the result of g(x). Since g(x)=x+2, f(g(x))=(x+2)^3. You substitute x+2 into the cube function. Function composition.
What is the end behavior of the function f(x)=-x^3?
As x??, f(x)? -?; as x?-?, f(x)??
As x??, f(x)?0; as x?-?, f(x)?0
As x??, f(x)??; as x?-?, f(x)?-?
As x??, f(x)?-?; as x?-?, f(x)?-?
A negative leading coefficient on an odd-degree polynomial means the graph falls to the right and rises to the left. Thus as x??, f(x)?-?, and as x?-?, f(x)??. This describes its end behavior. .
Find the x-intercepts of f(x)=x^3 - 4x.
No x-intercepts
x = -2, 0, and 2
x = 2 only
x = -4, 0, and 4
Set x^3 - 4x = x(x^2 -4) = 0. This factors to x(x-2)(x+2)=0, giving solutions x=-2, 0, and 2. Those are the x-intercepts. .
Determine the inflection point of f(x)=x^3 + 5.
(0, 0)
(0, 5)
(5, 0)
No inflection point
For f(x)=x^3+5, the second derivative f''(x)=6x. Setting this to zero gives x=0, which is the inflection point. Substituting in gives f(0)=5, so the point is (0,5). .
Which of the following is the equation of the function obtained by reflecting f(x)=x^3 across the y-axis, then shifting down 4 units, and then stretching vertically by a factor of 3?
-3x^3 + 12
3x^3 - 4
-3x^3 - 12
-x^3 - 12
Reflecting across the y-axis yields (-x)^3 = -x^3. Shifting down 4 gives -x^3 - 4. Stretching vertically by 3 multiplies the whole function by 3, resulting in -3x^3 - 12. .
After reflecting f(x)=x^3 across the x-axis, horizontally compressing by a factor of 1/2, and then shifting right 4 units, what is the resulting function?
-8(x - 4)^2
-8(x + 4)^3
8(x - 4)^3
-8(x - 4)^3
Reflecting across the x-axis gives -x^3. A horizontal compression by 1/2 replaces x with 2x, yielding -(2x)^3 = -8x^3. Finally shifting right 4 replaces x with (x-4), resulting in -8(x-4)^3. .
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Study Outcomes

  1. Understand Parent Function Fundamentals -

    Develop a clear definition for parent function, distinguishing core forms such as constant, linear, and cubic to build a solid conceptual base.

  2. Identify Linear Parent Functions -

    Recognize the parent function of a linear equation by matching its algebraic form to its graph and interpreting slope and intercept.

  3. Analyze Cubic Graph Parent Functions -

    Examine the shape and behavior of a cubic graph parent function, noting key features like inflection points and end behavior.

  4. Apply Transformations to Parent Functions -

    Use concepts from quiz 2-3 parent functions transformations graphing to perform shifts, reflections, stretches, and compressions on parent function graphs.

  5. Evaluate Graphing Skills through a Parent Functions Quiz -

    Test your mastery with a free parent functions quiz, reinforcing your ability to graph and interpret results accurately.

Cheat Sheet

  1. Understanding the Definition for Parent Function -

    The definition for parent function is the simplest form of a function that captures its basic shape, such as y=x for a line or y=x² for a parabola (source: Khan Academy). Remember "Base Form First" as a mnemonic to identify the original, untransformed equation before tackling a parent functions quiz. Recognizing this foundation helps you quickly classify and graph new functions.

  2. Identifying the Parent Function of a Linear Equation -

    In any parent function linear equation, y = x is your go-to model: it has a slope of 1 and no intercept shift (source: MIT OpenCourseWare). When asked "What is the parent function of a linear equation?" on a quiz, just check for f(x)=x or f(x)=mx with m=1 and b=0. This simplicity makes it perfect for mastering linear graph transformations in the parent functions quiz.

  3. Exploring Quadratic and Cubic Graph Parent Functions -

    Quadratic (y=x²) and cubic graph parent function (y=x³) are sisters in polynomial families, with parabolas and S-shaped curves respectively (source: Wolfram MathWorld). Note that y=x² is symmetric about the y-axis, while y=x³ has origin symmetry - this distinction is key in quiz 2-3 parent functions transformations graphing. Practice sketching both to see how coefficients affect steepness and reflection.

  4. Mastering Transformations in Quiz 2-3 Parent Functions -

    Transformations such as f(x - h)+k, a·f(x), and f( - x) shift, stretch, or reflect your base form; e.g., y=2(x - 3)²+4 moves the parabola right 3, up 4, and doubles its width (source: University of Arizona). Use the acronym "HST" (Horizontal, Stretch, Translation) to recall order of operations when graphing. Solid knowledge of these moves will skyrocket your confidence on any parent functions quiz.

  5. Graphing Constant Parent Functions -

    Constant graphs are the simplest: y=c creates a horizontal line at height c (source: Purplemath). These parent functions require no slope analysis, so jot "Zero Slope, Zero Stress" to remember that f(x)=5 is flat across all x. Quick mastery of constants makes you breeze through related quiz questions in seconds.

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