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Quizzes > Mathematics > Algebra

Identify Parent Functions and Their Graphs - Take the Quiz!

Think You Can Identify Which Parent Function a Graph Represents?

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
paper art illustration coral background free scored quiz parent functions graphs challenge yourself

Ready to master parent functions and their graphs? Our free, scored parent functions quiz challenges you to match each curve to its rule, perfect for anyone curious about what parent function is represented by the graph of y=x² or exploring a parent function of a radical function. You'll sharpen skills in identifying parent functions through quick, engaging questions. Whether you're prepping for exams or simply love interactive graphing, you'll find endless graphs practice to solidify your knowledge. Start now and see if you can ace every question!

Which parent function has a straight-line graph passing through the origin with a constant slope?
f(x) = x
f(x) = x^2
f(x) = |x|
f(x) = 1/x
The parent linear function f(x)=x is a straight line with slope 1 through the origin, exhibiting constant rate of change. Quadratic and absolute value produce curved shapes, and 1/x yields a hyperbola. Recognizing linear graphs is foundational in understanding function families. Linear Functions
Which parent function produces a U-shaped graph opening upward and symmetric about the y-axis?
f(x) = x^2
f(x) = ?x
f(x) = x^3
f(x) = |x|
The quadratic parent function f(x)=x^2 forms a U-shaped parabola symmetric about the y-axis. Square root is a half?line curve, cubic is S-shaped, and absolute value gives a V shape. Identifying parabolas is key for recognizing quadratic behavior. Quadratic Functions
Which parent function has a constant value for all x, resulting in a horizontal line?
f(x) = 1
f(x) = x
f(x) = x^2
f(x) = |x|
A constant function f(x)=c is represented by a horizontal line y=c that does not change with x. Unlike linear or quadratic functions, its rate of change is zero. Understanding constants helps in graphing and comparing different function behaviors. Constant Functions
Which parent function's graph forms a V shape with its vertex at the origin?
f(x) = |x|
f(x) = x^2
f(x) = ?x
f(x) = x^3
The absolute value function f(x)=|x| creates a V-shaped graph with its vertex at (0,0). It is piecewise linear and symmetric about the y-axis. This shape is distinct from parabolas and other curves. Absolute Value
Which parent function is only defined for x ? 0 and has a graph that starts at the origin and increases at a decreasing rate?
f(x) = ?x
f(x) = ?x
f(x) = x^2
f(x) = 1/x
The square root function f(x)=?x has domain [0,?) and grows slower as x increases, with slope decreasing. Cube root extends to negatives, quadratic grows faster, and reciprocal is undefined at zero. Recognizing domain restrictions is crucial. Square Root Function
Which parent function has a graph with two branches, one in the first quadrant and one in the third quadrant, and asymptotes at x = 0 and y = 0?
f(x) = 1/x
f(x) = e^x
f(x) = x^3
f(x) = ln(x)
The reciprocal function f(x)=1/x is a hyperbola with branches in QI and QIII, featuring vertical asymptote x=0 and horizontal asymptote y=0. Other options do not exhibit this two-branch asymptotic behavior. Reciprocal Function
Which parent function's graph is an S-shaped curve symmetric about the origin, increasing without bound in both directions?
f(x) = x^3
f(x) = x^2
f(x) = e^x
f(x) = |x|
The cubic function f(x)=x^3 is an odd function with an S-shaped graph, symmetric about the origin and unbounded as x?±?. Quadratic and absolute value have even symmetry, and exponential is one-sided. Cubic Function
Which parent function has a horizontal asymptote at y = 0, passes through (0,1), and increases rapidly as x increases?
f(x) = e^x
f(x) = ln(x)
f(x) = x
f(x) = 1/x
The exponential function f(x)=e^x crosses at (0,1), has y=0 as a horizontal asymptote for x?-?, and grows rapidly for positive x. Logarithm is undefined at x?0, and linear/reciprocal behave differently. Exponential Functions
Which parent function is the inverse of f(x) = e^x and has a vertical asymptote at x = 0?
f(x) = ln(x)
f(x) = 1/x
f(x) = x^3
f(x) = |x|
The natural logarithm f(x)=ln(x) is the inverse of the exponential function, defined for x>0, and features a vertical asymptote at x=0. It undoes the action of e^x. Logarithmic Functions
Which parent function has a graph that is always non-negative for all real x, is symmetric about the y-axis, and grows linearly?
f(x) = |x|
f(x) = x^2
f(x) = ?x
f(x) = x^3
Absolute value f(x)=|x| produces a V-shape, is non-negative, symmetric about the y-axis, and increases at a steady rate (linear). Quadratic grows quadratically, square root and cubic differ in domain/shape. Absolute Value
Which parent function has a graph that is defined for all real x, has an inflection point at the origin, and passes through (1,1) but increases at a decreasing rate for x>0?
f(x) = ?x
f(x) = x^3
f(x) = ?x
f(x) = ln(x)
The cube root function f(x)=?x is defined for all real x, passes through the origin as an inflection point, and grows with decreasing slope for positive x. It is the inverse of the cubic function. Cube Root Function
Identify the parent function used in f(x) = ?(x - 4) + 1.
f(x) = ?x
f(x) = 1/x
f(x) = x^2
f(x) = ln(x)
The given function is a horizontal shift right by 4 and up by 1 of the square root parent f(x)=?x. Recognizing shifts helps identify the underlying parent function. Function Transformations
Which parent function is neither even nor odd, showing no symmetry about the axes or origin?
f(x) = e^x
f(x) = x^2
f(x) = x^3
f(x) = |x|
The exponential function f(x)=e^x has no symmetry about the y-axis or origin, making it neither even nor odd. Polynomials of even or odd degree and absolute value have specific symmetries. Function Symmetry
Which parent function has a graph that is discontinuous at every integer value of x?
f(x) = ?x?
f(x) = x^3
f(x) = ln(x)
f(x) = ?x
The greatest integer or floor function f(x)=?x? jumps at each integer, causing discontinuities there. Other listed functions are continuous on their domains. Greatest Integer Function
Which parent function’s range is (0, ?) and is defined for all real x?
f(x) = e^x
f(x) = ln(x)
f(x) = ?x
f(x) = 1/x
Exponential functions f(x)=e^x are defined for all real x and yield only positive outputs, giving a range of (0,?). Logarithms and square roots have different domains, and reciprocal can be negative. Exponential Functions
Which parent function has an inflection point at the origin and is the simplest odd-degree polynomial beyond linear?
f(x) = x^3
f(x) = x
f(x) = x^5
f(x) = x^2
The cubic function f(x)=x^3 is the next simplest odd polynomial after f(x)=x, featuring an inflection at the origin. Linear x has no inflection, and higher odds are more complex. Inflection Point
Which parent function’s graph has a point of inflection at the origin?
f(x) = x^3
f(x) = x^2
f(x) = |x|
f(x) = ?x
An inflection point is where concavity changes. The cubic function f(x)=x^3 changes concavity at the origin. Quadratic and absolute value do not, and ?x lacks negative domain. Inflection Point
Which parent function is the inverse of f(x) = x^3?
f(x) = ?x
f(x) = x^2
f(x) = ln(x)
f(x) = 1/x
The cube root function f(x)=?x is the inverse of the cubic function, swapping x and y such that (y)^3 = x. Quadratic, logarithm, and reciprocal are inverses of different functions. Cube Root Function
Which parent function has a constant second derivative for all x?
f(x) = x^2
f(x) = x^3
f(x) = x
f(x) = e^x
For f(x)=x^2, the second derivative is f''(x)=2, a constant for all x. Cubic and linear yield variable or zero second derivatives, and exponential’s second derivative matches f(x), not constant. Derivatives FAQ
Which parent function is both one-to-one and onto the set of all real numbers?
f(x) = x
f(x) = x^2
f(x) = |x|
f(x) = 1/x
The identity function f(x)=x pairs each real x with itself, covering all real outputs with no repeats. Quadratic and absolute value are not one-to-one, and reciprocal omits zero. One-to-One Functions
Which parent function’s graph is symmetric with respect to the origin and has both vertical and horizontal asymptotes?
f(x) = 1/x
f(x) = x^3
f(x) = ln(x)
f(x) = e^x
The reciprocal function f(x)=1/x is odd (origin symmetric) and has vertical asymptote x=0 and horizontal asymptote y=0. Cubic is odd but has no asymptotes, and logarithm/exponential have different behavior. Reciprocal Function
Which parent function is its own derivative, meaning f?'(x) = f(x)?
f(x) = e^x
f(x) = x^2
f(x) = ln(x)
f(x) = 1/x
The exponential function f(x)=e^x uniquely satisfies f?'(x)=e^x = f(x). Polynomial and logarithmic functions have different derivative forms, and reciprocal yields negative exponent derivative. Exponential Derivatives
Which parent function has a vertical tangent (infinite slope) at x = 0+ and is only defined for x ? 0?
f(x) = ?x
f(x) = ln(x)
f(x) = x^2
f(x) = |x|
At x=0+, the slope of f(x)=?x approaches infinity, producing a vertical tangent. It is defined only for x?0. Logarithm is undefined at 0, and quadratic/absolute value have finite slopes. Square Root Tangent
Which parent function has a domain of x > 0 and range of all real numbers?
f(x) = ln(x)
f(x) = e^x
f(x) = ?x
f(x) = 1/x
The natural logarithm f(x)=ln(x) is defined only for positive x and can output any real value. Exponential outputs only positives, square root only non-negatives, and reciprocal omits zero in range. Logarithmic Functions
Which parent function among these exhibits oscillatory behavior with a period of 2? and a range of [?1,1]?
f(x) = sin(x)
f(x) = e^x
f(x) = x^3
f(x) = ln(x)
The sine function f(x)=sin(x) oscillates between ?1 and 1 with a fundamental period of 2?. Other options are non?oscillatory and unbounded or have different domains. Understanding trigonometric parent functions is key in advanced graph analysis. Trigonometric Functions
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Study Outcomes

  1. Identify Common Parent Functions -

    After completing the quiz, you will recognize the graphs of fundamental parent functions such as linear, quadratic, cubic, absolute value, and square root.

  2. Match Graphs to Parent Functions -

    You will be able to determine what parent function is represented by the graph by analyzing key features like shape, intercepts, and symmetry.

  3. Analyze Transformations -

    You will discern how shifts, reflections, stretches, and compressions affect the graphs of parent functions in the quiz's 2-3 parent functions transformations graphing section.

  4. Recognize Radical Function Graphs -

    You will identify the parent function of a radical function by spotting the distinctive characteristics of square root and cube root graphs.

  5. Differentiate Exponential and Logarithmic Graphs -

    You will distinguish between the parent functions of exponential and logarithmic families by their growth, decay, and asymptotic behavior.

  6. Apply Graphing Skills in a Scored Quiz -

    You will confidently apply your understanding of parent functions and their graphs to answer scored quiz questions and track your mastery.

Cheat Sheet

  1. Linear Parent Function (f(x)=x) -

    The graph of the linear parent function is a straight line through the origin with slope 1, representing a constant rate of change for all real x. In slope-intercept form y=mx+b, setting m=1 and b=0 simplifies to f(x)=x, making it the simplest parent functions and their graphs model (Source: Khan Academy).

  2. Quadratic Parent Function (f(x)=x²) -

    This classic U-shaped parabola has its vertex at (0,0) and is symmetric about the y-axis, making it one of the most recognizable parent functions and their graphs. Remember "U for U-shape" as a mnemonic, and note that f(x)=x² models area growth and appears in physics and finance contexts (Source: MIT OpenCourseWare).

  3. Absolute Value Parent Function (f(x)=|x|) -

    The V-shaped graph of f(x)=|x| reflects negative inputs to the right side, producing two linear arms with slopes ±1 and range y≥0. Use the trick "V for Value" to quickly identify this parent function of a radical function in quizzes and homework (Source: University of Texas at Austin).

  4. Square Root Parent Function (f(x)=√x) -

    This radical function starts at the origin and curves upward for x≥0, mapping inputs to their principal square roots and illustrating limited domain behavior. When considering "what parent function is represented by the graph" of a radical, the half-parabola of √x is the key shape to memorize (Source: Purplemath).

  5. Exponential & Logarithmic Parent Functions (f(x)=bˣ and f(x)=log_b x) -

    The exponential curve y=bˣ (b>1) rises rapidly and passes through (0,1), while its inverse log graph y=log_b x climbs slowly and passes through (1,0). Recall "exponents shoot up, logs climb up" as a simple way to distinguish these two essential parent functions and their graphs (Source: Wolfram MathWorld).

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