Identify zeros and intercepts of a function

A zero is an input in a function’s domain for which $f(x)=0$ ; each real zero corresponds to an $x$ -intercept $(x,0)$ , while the $y$ -intercept is $(0,f(0))$ only when $0$ belongs to the domain. These relationships can be identified from equations, tables, and graphs, including cases with no real zeros; the scope is limited to real zeros and intercepts, not complex zeros or advanced multiplicity analysis.

Detailed Explanation: 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.

Learn by doing: Identify zeros and intercepts of a function

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Linear Equations - Find Y Intercept (Integer) - Standard Form

Earned ?