Lesson 5.3 : Graphs of Polynomial Functions

New Concept

Discover how a polynomial's formula dictates its graph. We'll connect its degree, factors, and leading term to key visual features like zeros, end behavior, and turning points, empowering you to sketch and interpret polynomial functions.

What’s next

Now, let's dive into practice cards on finding zeros and multiplicities. Soon, you’ll apply these skills in interactive examples to sketch complete polynomial graphs.

Property

Recall that if $f$ is a polynomial function, the values of $x$ for which $f(x) = 0$ are called zeros of $f$. Given a polynomial function $f$, find the $x$-intercepts by factoring.

1. Set $f(x) = 0$.
2. If the polynomial function is not given in factored form:

a. Factor out any common monomial factors.
b. Factor any factorable binomials or trinomials.

3. Set each factor equal to zero and solve to find the $x$-intercepts.

Examples

• To find the x-intercepts of $f(x) = x^4 - 5x^3 + 4x^2$, we factor to get $x^2(x-1)(x-4)=0$. The intercepts are at $(0,0)$, $(1,0)$, and $(4,0)$.
• To find the x-intercepts of $f(x) = x^3 - 3x^2 - 4x + 12$, we factor by grouping: $x^2(x-3) - 4(x-3) = (x^2-4)(x-3)=0$. The intercepts are at $(2,0)$, $(-2,0)$, and $(3,0)$.
• For $g(x) = (x-5)^2(2x+1)$, the factors are already found. Setting them to zero gives $x=5$ and $x = -\frac{1}{2}$. The intercepts are $(5,0)$ and $(-\frac{1}{2}, 0)$.

Explanation

Zeros are where the graph hits the x-axis. Factoring breaks the polynomial into small pieces. Setting each piece to zero reveals the x-intercepts, giving you key points to start sketching the graph.

Property

The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity. If a polynomial contains a factor of the form $(x - h)^p$, the behavior near the $x$-intercept $h$ is determined by the power $p$. We say that $x = h$ is a zero of multiplicity $p$.

Examples

• The function $f(x)=(x-3)^4(x+1)^3$ has a zero of multiplicity 4 at $x=3$, where the graph touches the axis and bounces off. It has a zero of multiplicity 3 at $x=-1$, where the graph crosses the axis.
• If a graph crosses the x-axis at $x=-2$ and touches the x-axis at $x=4$, a possible formula has factors $(x+2)$ with an odd power and $(x-4)^2$ with an even power.
• Near its zero at $x=5$, the function $f(x) = (x-5)^3$ flattens as it crosses the axis, similar to the behavior of the parent cubic function $y=x^3$ at the origin.

Explanation

The graph of a polynomial function will touch the $x$-axis at zeros with even multiplicities. The graph will cross the $x$-axis at zeros with odd multiplicities.
The sum of the multiplicities is the degree of the polynomial function.
Multiplicity tells you how the graph behaves at a zero. An odd multiplicity means the graph crosses the x-axis. An even multiplicity means the graph touches the axis and "bounces" back, changing direction without crossing.

Property

The behavior of a graph of a polynomial function of the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$ will either ultimately rise or fall as $x$ increases without bound and will either rise or fall as $x$ decreases without bound. This is called the end behavior. It is determined by the leading term, $a_n x^n$.

• If $n$ is even and $a_n > 0$, the graph rises on both ends.
• If $n$ is even and $a_n < 0$, the graph falls on both ends.
• If $n$ is odd and $a_n > 0$, the graph falls on the left and rises on the right.
• If $n$ is odd and $a_n < 0$, the graph rises on the left and falls on the right.

Examples

• For $f(x) = -4x^5 + x^3 - 1$, the degree is 5 (odd) and the leading coefficient is negative. The graph rises on the left ($x \to -\infty, f(x) \to \infty$) and falls on the right ($x \to \infty, f(x) \to -\infty$).
• For $g(x) = 3x^4 - 2x + 7$, the degree is 4 (even) and the leading coefficient is positive. The graph rises on both the left and the right ($x \to \pm\infty, f(x) \to \infty$).
• The function $h(x) = (2-x)(x+3)^2$ has a leading term of $-x^3$. Since the degree is odd and the coefficient is negative, it rises on the left and falls on the right.

Explanation

Think about what happens when x gets huge. The term with the highest power takes over and dictates whether the graph's arms point up or down. This gives you the big picture of the graph's direction.

Property

A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). It is a point where a local maximum or local minimum occurs.
A polynomial of degree $n$ will have at most $n - 1$ turning points.

Examples

• The polynomial $f(x) = 2x^7 - x^3 + 5$ has a degree of 7. Therefore, it can have at most $7-1=6$ turning points.
• Consider $g(x) = (x-3)(x+1)(x-5)^2$. The degree is $1+1+2=4$. The maximum number of turning points for this function is $4-1=3$.
• If a polynomial graph has 4 turning points, its degree must be at least 5. A degree of 4 would allow for a maximum of only 3 turning points.

Explanation

A polynomial cannot have more turns than its degree minus one. A degree 5 polynomial can change direction from rising to falling (or vice versa) at most 4 times. This rule helps you check if your sketch is reasonable.

Property

Let $f$ be a polynomial function. The Intermediate Value Theorem states that if $f(a)$ and $f(b)$ have opposite signs, then there exists at least one value $c$ between $a$ and $b$ for which $f(c) = 0$.

Examples

• To show that $f(x) = x^3 + x - 5$ has a zero between $x=1$ and $x=2$, we check the outputs. Since $f(1)=-3$ (negative) and $f(2)=5$ (positive), a zero must exist between 1 and 2.
• To show that $g(x) = x^4 - 6x - 8$ has a zero between $x=2$ and $x=3$, we find $g(2)=-4$ and $g(3)=55$. Because the signs are opposite, there is a zero between 2 and 3.
• Consider $h(x) = x^2+1$ on the interval $[-1, 1]$. We find $h(-1)=2$ and $h(1)=2$. Since the signs are the same, the theorem does not guarantee a zero in this interval.

Explanation

If a continuous function's graph is at a positive height at one point and a negative height at another, it must cross the x-axis to get from one to the other. This theorem guarantees a zero exists in that interval.

Property

If a polynomial of lowest degree $p$ has horizontal intercepts at $x = x_1, x_2, \ldots, x_n$, then the polynomial can be written in the factored form: $f(x) = a(x - x_1)^{p_1}(x - x_2)^{p_2} \cdots (x - x_n)^{p_n}$ where the powers $p_i$ on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor $a$ can be determined given a value of the function other than the $x$-intercept.

Examples

• A graph has zeros at $x=-2$ (crosses) and $x=3$ (bounces), and a y-intercept at $(0,-18)$. The form is $f(x)=a(x+2)(x-3)^2$. Using $(0,-18)$, we find $-18=a(2)(-3)^2$, so $a=-1$. The formula is $f(x)=-(x+2)(x-3)^2$.
• A graph crosses linearly at $x=-1$ and $x=1$, and passes through $(2, 9)$. The form is $f(x)=a(x+1)(x-1)$. Using $(2,9)$, we find $9=a(3)(1)$, so $a=3$. The formula is $f(x)=3(x+1)(x-1)$.
• A graph has a zero at $x=0$ that flattens like a cubic and a zero at $x=4$ that crosses linearly. It passes through $(-1, -5)$. The form is $f(x)=ax^3(x-4)$. Using $(-1, -5)$, we find $-5=a(-1)^3(-5)$, so $a=-1$. The formula is $f(x)=-x^3(x-4)$.

Explanation

You can build a polynomial's formula from its graph. Each x-intercept gives a factor. The graph's behavior (crossing or bouncing) gives the factor's exponent (multiplicity). Use one other point to find the stretch factor 'a'.