Graphs of Polynomial Functions – Practice Questions with Answers
Practice questions and answers on graphs of polynomial functions are presented below. These questions can be used as a self-test or classroom practice. If needed, you may use free graph paper.
The questions focus on factoring polynomials, finding the zeros of a polynomial, and graphing a polynomial function using sign tables and the leading coefficient rule.
How to use this test: Try each question on your own first, writing down your reasoning. Then open the solution to check your work and read the detailed explanation. Each question links to a page with the candidate graphs so you can compare them visually.
Before You Begin: Strategy for Identifying Polynomial Graphs
To match a polynomial equation to its graph, work through these steps in order:
- Degree and leading coefficient: Use these to determine end behavior (the leading coefficient rule).
- Factor: Fully factor the polynomial to reveal all real zeros.
- Zeros and multiplicity: Each real zero is an x-intercept. Odd multiplicity → graph crosses the axis. Even multiplicity → graph touches the axis and turns.
- y-intercept: Evaluate \( f(0) \).
- Sign table: Test intervals between zeros to see where \( f(x) \) is positive or negative.
- Turning points: A degree-\( n \) polynomial has at most \( n-1 \) turning points.
Practice Test
Question 1
Identify the graph of the polynomial function \( f \).
\[ f(x) = x - 1 \]View the candidate graphs for Question 1 →
Solution to Question 1
Answer: d
Explanation: This is a linear function (degree 1) with a slope of \( 1 \) and a y-intercept of \( -1 \). The graph is a straight line rising from left to right, crossing the x-axis at \( (1, 0) \) and the y-axis at \( (0, -1) \).
Question 2
Identify the graph of the polynomial function \( f \).
\[ f(x) = x^2 + 2x - 3 \]View the candidate graphs for Question 2 →
Solution to Question 2
Answer: a
Explanation: This is a quadratic function (degree 2, positive leading coefficient) opening upwards. Factoring: \[ f(x) = (x+3)(x-1) \] so the graph has x-intercepts at \( x = -3 \) and \( x = 1 \). The y-intercept is \( f(0) = -3 \). The vertex is at \( x = -1 \), \( y = -4 \).
Question 3
Identify the graph of the polynomial function \( f \).
\[ f(x) = x^2 - 3 \]View the candidate graphs for Question 3 →
Solution to Question 3
Answer: d
Explanation: This is a quadratic function opening upwards with its vertex (and y-intercept) at \( (0, -3) \). Its roots are at \( x = \pm\sqrt{3} \approx \pm 1.732 \). Since the vertex is below the x-axis and the parabola opens upward, it crosses the x-axis twice.
Question 4
Identify the graph of the polynomial function \( f \).
\[ f(x) = x^3 - 2x^2 - x + 2 \]View the candidate graphs for Question 4 →
Solution to Question 4
Answer: a
Explanation: This cubic function (degree 3, positive leading coefficient: left end down, right end up) can be factored by grouping: \[ x^2(x-2) - 1(x-2) = (x^2-1)(x-2) = (x-1)(x+1)(x-2) \] The graph crosses the x-axis at \( x = -1, 1, \) and \( 2 \) (all odd multiplicity). The y-intercept is \( f(0) = 2 \). Up to 2 turning points are possible.
Question 5
Identify the graph of the polynomial function \( f \).
\[ f(x) = x^3 - x^2 + x - 1 \]View the candidate graphs for Question 5 →
Solution to Question 5
Answer: d
Explanation: Factoring by grouping gives: \[ x^2(x-1) + 1(x-1) = (x^2+1)(x-1) \] Since \( x^2 + 1 = 0 \) has no real solutions, the only real root is \( x = 1 \). The graph crosses the x-axis once, at \( (1, 0) \), and has a y-intercept of \( f(0) = -1 \).
Question 6
Identify the graph of the polynomial function \( f \).
\[ f(x) = x^4 + x^3 - x^2 + x - 2 \]View the candidate graphs for Question 6 →
Solution to Question 6
Answer: a
Explanation: This is a quartic even function (degree 4, positive leading coefficient), so both ends go up. The y-intercept is \( f(0) = -2 \). By testing rational roots, we find that \( x = 1 \) and \( x = -2 \) are roots. The graph will have at most 3 turning points.
Question 7
Identify the graph of the polynomial function \( f \).
\[ f(x) = x^4 - 2x^3 - x^2 + 2x \]View the candidate graphs for Question 7 →
Solution to Question 7
Answer: c
Explanation: Factor out an \( x \): \[ x(x^3 - 2x^2 - x + 2) \] Using the grouping from Question 4: \[ x(x-1)(x+1)(x-2) \] The graph crosses the x-axis at \( -1, 0, 1, \) and \( 2 \) (all multiplicity 1). Both ends go up because the degree is even and the leading coefficient is positive.
Question 8
Identify the graph of the polynomial function \( f \).
\[ f(x) = x^4 + 1 \]View the candidate graphs for Question 8 →
Solution to Question 8
Answer: c
Explanation: This is the base quartic function shifted up by \( 1 \). It opens upwards, has a minimum (and y-intercept) at \( (0, 1) \), and has no real roots since \( x^4 + 1 > 0 \) for all real numbers. The graph never crosses the x-axis.
Question 9
Identify the graph of the polynomial function \( f \).
\[ f(x) = x^4 - 2 \]View the candidate graphs for Question 9 →
Solution to Question 9
Answer: a
Explanation: This is a quartic function shifted down by \( 2 \). It opens upwards, has a minimum (and y-intercept) at \( (0, -2) \), and has real roots at \( x = \pm\sqrt[4]{2} \approx \pm 1.189 \).
Question 10
Identify the graph of the polynomial function \( f \).
\[ f(x) = x^4 + x^3 - 2x^2 \]View the candidate graphs for Question 10 →
Solution to Question 10
Answer: c
Explanation: Factor out \( x^2 \): \[ x^2(x^2 + x - 2) = x^2(x+2)(x-1) \] The graph touches and turns at the x-axis at \( x = 0 \) (multiplicity 2, even) and crosses it at \( x = -2 \) and \( x = 1 \) (multiplicity 1, odd). Both ends go up because the degree is even and the leading coefficient is positive.
Frequently Asked Questions
How do you identify the graph of a polynomial function?
Factor the polynomial to find its real zeros (x-intercepts), determine the multiplicity of each zero to know whether the graph crosses or touches the x-axis, use the leading coefficient rule to determine end behavior, and build a sign table to know where the graph is above or below the x-axis.
What is the leading coefficient rule for polynomial graphs?
For even-degree polynomials, a positive leading coefficient means both ends go up; a negative leading coefficient means both ends go down. For odd-degree polynomials, a positive leading coefficient means the left end goes down and the right end goes up; negative reverses this.
How do you find the zeros of a polynomial?
Set the polynomial equal to zero and solve. Factoring (by grouping, using the quadratic formula, or by testing rational roots) is the most common method. Each real zero corresponds to an x-intercept.
What is multiplicity and how does it affect the graph?
Multiplicity is the number of times a factor \( (x - r) \) appears in the factored form. If the multiplicity is odd, the graph crosses the x-axis at \( r \). If the multiplicity is even, the graph touches the x-axis at \( r \) and turns around.
How many turning points can a polynomial have?
A polynomial of degree \( n \) can have at most \( n - 1 \) turning points. For example, a cubic (degree 3) can have up to 2 turning points, and a quartic (degree 4) can have up to 3.