Skill: Identify zeros and intercepts of a function

A zero of a function is an input value $x$ that makes the function equal to $0$:

$$f(x)=0$$

Each real zero gives an $x$-intercept:

$$(x,0)$$

To find the $y$-intercept, substitute $(x=0)$. The point is

$$(0,f(0)),$$

as long as $0$ is in the function’s domain.

Example

Find the zeros, $x$-intercepts, and $y$-intercept of

$$f(x)=x^2-3x-4.$$

Step 1: Find the zeros

Set the function equal to zero:

$$x^2-3x-4=0.$$

Factor:

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

Set each factor equal to zero:

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

So the zeros are

$$x=4 \quad \text{and} \quad x=-1.$$

Step 2: Find the $x$-intercepts

Use each zero as the $x$-coordinate and pair it with $0$:

$$(4,0) \quad \text{and} \quad (-1,0).$$

Therefore, the $x$-intercepts are

$$\boxed{(4,0)\text{ and }(-1,0)}.$$

Step 3: Find the $y$-intercept

Substitute $(x=0)$:

$$f(0)=0^2-3(0)-4=-4.$$

Therefore, the $y$-intercept is

$$\boxed{(0,-4)}.$$

For this polynomial, every real number is in the domain, so $(x=0)$ is allowed. In general, if $0$ is not in the domain, the function has no $y$-intercept.