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# Finding the Y-Intercept of a Function

(KS4, Year 10)

## 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**?

## Step-by-Step:

# 1

# 2

Substitute

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

## Answer:

The y-intercept of**f(x) = 2x − 3**is

**−3**.

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

## Question

What is the y-intercept of the function**f(x) = x**?

^{2}+ 4x + 2## Step-by-Step:

# 1

Write the function. This is a quadratic equation.

f(x) = x

^{2}+ 4x + 2**Note:**x^{2}= x × x (x squared)**and**4x = 4 × x.# 2

Substitute

**x = 0**into the function.
f(0) = 0

^{2}+ 4 × 0 + 2 f(0) = 0 × 0 + 4 × 0 + 2 f(0) = 0 + 0 + 2 f(0) = 2## Answer:

The y-intercept of**f(x) = x**is

^{2}+ 4x + 2**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

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