MA001: College Algebra

In addition to the end behavior of polynomial functions, we are also interested in what happens in the "middle" of the function. In particular, we are interested in locations where graph behavior changes. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing.

We are also interested in the intercepts. As with all functions, the $y$-intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one $y$-intercept $\left(0, a_{0}\right)$. The $x$-intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one $x-$intercept. See Figure 9.

Figure 9

INTERCEPTS AND TURNING POINTS OF POLYNOMIAL FUNCTIONS

A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The $y$-intercept is the point at which the function has an input value of zero. The $x$-intercepts are the points at which the output value is zero.

HOW TO

Given a polynomial function, determine the intercepts.

EXAMPLE 8

Determining the Intercepts of a Polynomial Function

Given the polynomial function $f(x)=(x-2)(x+1)(x-4)$, written in factored form for your convenience, determine the y- and x-intercepts.

Solution

The $y$-intercept occurs when the input is zero so substitute $0$ for $x$.

$\begin{aligned} f(0) &=f(0)=(0-2)(0+1)(0-4) \\ &=(-2)(1)(-4) \\ &=8 \end{aligned}$

The $y$-intercept is $(0, 8)$.

The $x$-intercepts occur when the output is zero.

$0=(x-2)(x+1)(x-4)$

$x-2=0 \quad \text { or } \quad x+1=0 \quad \text { or } \quad x-4=0$

$x=2 \quad \text { or } \quad x=-1 \quad \text { or } \quad x=4$

The $x$-intercepts are $(2,0)$, $(–1,0)$, and $(4,0)$.

We can see these intercepts on the graph of the function shown in Figure 10.

Figure 10

EXAMPLE 9

Determining the Intercepts of a Polynomial Function with Factoring

Given the polynomial function $f(x)=x^{4}-4 x^{2}-45$, determine the $y$- and $x$-intercepts.

Solution

The $y$-intercept occurs when the input is zero.

$\begin{aligned} f(0) &=(0)^{4}-4(0)^{2}-45 \\ &=-45 \end{aligned}$

The $y$-intercept is $(0,−45)$.

The $x$-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.

$\begin{aligned} f(x) &=x^{4}-4 x^{2}-45 \\ &=\left(x^{2}-9\right)\left(x^{2}+5\right) \\ &=(x-3)(x+3)\left(x^{2}+5\right) \end{aligned}$

$0=(x-3)(x+3)\left(x^{2}+5\right)$

$x-3=0 \quad \text { or } \quad x+3=0 \quad \text { or } \quad x^{2}+5=0$

$x=3 \quad \text { or } \quad x=-3 \quad \text { or } \quad \text { (no real solution) }$

The $x$-intercepts are $(3,0)$ and $(–3,0)$.

We can see these intercepts on the graph of the function shown in Figure 11. We can see that the function is even because $f(x)=f(-x)$.

Figure 11

TRY IT #6

Given the polynomial function $f(x)=2 x^{3}-6 x^{2}-20 x$, determine the $y$- and $x$-intercepts.