## The Lesson

The y-intercept of a function is found where the graph of a function crosses the y-axis on a pair of Cartesian coordinate axes.

## How to Find the Y-intercept of a Function

The y-intercept of a function f(x) is found by evaluating the function at x = 0.

This uses functional notation (see Note) to express the following idea. The y-intercept is the value of the function at the y-axis. Along the y-axis, the value of x is 0. The input x of the function equals 0. The image below shows what we mean:

## A Real Example of How to Find the Y-intercept of a Function

Finding the y-intercept of a function is easy.

## Question

What is the y-intercept of the function f(x) = 2x − 3?

# 1

Write the function. This is a linear equation.
f(x) = 2x − 3
Note: 2x = 2 × x.

# 2

Substitute x = 0 into the function.
f(0) = 2 × 0 − 3 f(0) = 0 − 3 f(0) = −3

The y-intercept of f(x) = 2x − 3 is −3.

## Lesson Slides

The slider below shows another real example of how to find the y-intercept of a function. Open the slider in a new tab

## Question

What is the y-intercept of the function f(x) = x2 + 4x + 2?

# 1

Write the function. This is a quadratic equation.
f(x) = x2 + 4x + 2
Note: x2 = x × x (x squared) and 4x = 4 × x.

# 2

Substitute x = 0 into the function.
f(0) = 02 + 4 × 0 + 2 f(0) = 0 × 0 + 4 × 0 + 2 f(0) = 0 + 0 + 2 f(0) = 2

The y-intercept of f(x) = x2 + 4x + 2 is 2.

## A Note on Notation

For a function, the input is often denoted by x and the function by f(x), which is equal to the output.

By choosing an input x, the function gives an output. To find the y-intercept, the input x = 0. The function f(x) becomes f(0).

The notation is the same for any value of x, whether a number or symbol:

The key point is that x is a variable. It can take many values. The function tells you how to relate that input to an output. When a certain value of x is substituted in, a certain output comes out.

## Evaluating a Function

Evaluating a function means putting an input into a function, and finding the output it relates to. When a value is chosen for the input x, the output can be found by substituting that value in for x wherever it is found in the function. For example, if the function is:
f(x) = x + 1
Evaluating the function at x = 0 means substituting 0 for x wherever it is found in the function.
f(0) = 0 + 1 f(0) = 1