# How to find y intercept

Here you will learn about how to find the y intercept from a straight line graph, including straight lines in the slope intercept form, y=mx+b and standard form, ax+by=c.

Students will first learn about how to find the y intercept in 8 th grade math with their work with linear functions.

## What is the y intercept and how do you find the y-intercept?

Finding the y intercept of a straight line is an important skill used to solve algebraic and real-life problems involving straight line graphs.

The intercepts of a graph are where the graph crosses the coordinate axes.

You can find the y intercepts of graphs of all types of functions including straight lines, quadratic functions, cubic functions and others. In this case, the y intercept is the point where the function crosses or intercepts the y -axis.

To find the y intercept(s) of a line:

Substitute x=0 into the equation of the function and evaluate for y.

To find the x intercept(s) of a line:

Substitute y=0 into the equation of the function and evaluate for x.

Note, this is a useful strategy to draw the equation of a straight line.

Step-by-step guide: Graphing Linear Equations

Example

The y intercept of a straight line is the value of y when the x -coordinate is zero.

In this real life example, the y intercept represents the starting value or fixed price, with the slope or gradient representing the unit rate or rate of change.

Let’s look at some examples.

A straight line has the equation, in standard form, 4x-3y=18. Find the y intercept of the line.

To find the y intercept, substitute x=0 into the equation.

4(0)-3y=18

Now solve the equation to find the corresponding y value.

\begin{aligned}-3y&=18 \\\\ y&=-6 \end{aligned}

The y intercept is -6 and has coordinate (0,-6).

Note: If any function is of the form y=f(x)+\text{constant}, the constant is the y intercept. This is because the y -intercept is always when the x -value is 0, so when substituting 0 for x the constant is going to be the y -value.

See examples of different types of polynomials below.

## Common Core State Standards

How does this relate to high school math?

• High School: Functions (HS.IF.C.7a)
Graph linear and quadratic functions and show intercepts, maxima, and minima.

## How to find the y intercept

In order to find the y intercept, you need to:

1. Substitute \textbf{x} = \bf{0} into the equation of the line.
2. Solve the equation for \textbf{y}.

## How to find y intercept examples

### Example 1: finding the y intercept of a line in the form y = mx + b

Find the y intercept of the line y=2x-5.

1. Substitute \textbf{x} = \bf{0} into the equation of the line.

y=2(0)-5

2Solve the equation for \textbf{y}.

This equation gives y=-5.

The y intercept is -5.

It has coordinates (0,-5).

Note: This is the special case where the equation is in the form y = [function of x ] + \; c (constant). In this case c=-5 so the y intercept is (0,-5).

### Example 2: finding the y intercept of a line in the form y = mx + b

Find the y intercept of the line y=\cfrac{1}{2}x+3

Substitute \textbf{x} = \bf{0} into the equation of the line.

Solve the equation for \textbf{y}.

### Example 3: finding the y intercept of a line in the form y = mx + b

Find the y intercept of the line y=9x-14.

Substitute \textbf{x} = \bf{0} into the equation of the line.

Solve the equation for \textbf{y}.

### Example 4: finding the y intercept of a line in the form ax + by = c

Find the y intercept of the line 2x+5y=20.

Substitute \textbf{x} = \bf{0} into the equation of the line.

Solve the equation for \textbf{y}.

### Example 5: finding the y intercept of a line in the form ax + by = c

Find the y intercept of the line 3x-4y=24.

Substitute \textbf{x} = \bf{0} into the equation of the line.

Solve the equation for \textbf{y}.

### Example 6: finding the y intercept of a line in the form ax + by = c

Find the y intercept of the line 7x+9y=36.

Substitute \textbf{x} = \bf{0} into the equation of the line.

Solve the equation for \textbf{y}.

### Teaching tips for how to find y intercept

• While introducing the concept of the y -intercept, use multiple visuals where students can clearly see that the y -intercept is the point where the line crosses the y -axis.

• The use of interactive activities, such as online graphing tools or activities that require students to physically graph activities, are just as effective as worksheets with practice problems.

• For students that are struggling, break down the process of finding the y -intercept into simple steps. The use of visuals for each step can aid students in understanding the process.

### Easy mistakes to make

• Substituting the wrong value for zero
A common error is to think that the y intercept is when y=0. It is important to remember that when y=0, the line will be crossing the x -axis and therefore will give the x intercept.

• Forgetting to state the coordinates when asked
If an exam question asks for the y intercept or x intercept, just the value where the line crosses the axes is appropriate. However, sometimes the question will ask for the coordinates of the y intercept. In this case the answer must be given in the form (0,b) for the y intercept, or (-\cfrac{b}{m},0) for the coordinates of the x intercept.

• Confusing the intercept with the gradient
When the line is in the form y=mx+b, a common error is to confuse the y intercept, b with the gradient or slope of a line, m. The y intercept will have coordinates (0,b) and the x intercept will have coordinates (-\cfrac{b}{m},0).

### Practice how to find the y intercept questions

1. State the coordinate of the y intercept of the line y=3x-2.

(-2,0)

(0,-2)

(3,0)

(0,3)

Substitute x=0 to give

\begin{aligned}y&=3(0)-2 \\\\ y&=-2 \end{aligned}

The coordinate of the y -intercept is (0,-2).

2. Find the coordinate of the y intercept of the line y=3x-6.

(0,-6)

(0,2)

(2,0)

(-6,0)

Substitute x=0 to give

\begin{aligned}y&=3(0)-6 \\ y&=-6 \end{aligned}

The coordinate of the y -intercept is (0,-6).

3. The equation of a line is given as y=8-3x. Find the y intercept.

\cfrac{8}{3}

-3

-8

8

When x=0,

\begin{aligned}y&=8-3(0) \\\\ y&=8\end{aligned}

4. The equation of a line is given as x-3y=9. Find the y intercept.

-3

3

-\cfrac{1}{3}

9

When x=0,

\begin{aligned}-3y&=9 \\\\ y&=-3 \end{aligned}

5. The equation of a line is given as 5x-4y=10. Find the y intercept.

2

-2.5

-\cfrac{2}{5}

\cfrac{1}{2}

When x=0,

\begin{aligned}5(0)-4y&=10 \\\\ -4y&=10 \\\\ y&=-2.5 \end{aligned}

6. The equation of a line is given as -2x+12y=6. Find the y intercept.

\cfrac{1}{2}

\cfrac{1}{6}

-2

3

When x=0,

\begin{aligned}-2(0)+12y&=6 \\\\ 12y&=6 \\\\ y&=\cfrac{6}{12}=\cfrac{1}{2} \end{aligned}

## How to find y intercept FAQs

How do I find the \textbf{x} -intercept of a line?

The process for finding the x -intercept of a line is similar to finding the y -intercept. Instead of substituting x=0, you would substitute y=0, and solve the equation for x.

Can you find the \textbf{y} -intercept when the equation is in point-slope form?

Yes, the process for finding the y -intercept in point-slope form is similar to any other linear equation. You will substitute x=0 into the equation and then solve for y.

How do you find the slope of the line?

To find the slope of the line, you will use the formula: m=\cfrac{y_{2}-y_{1}}{x_{2}-x_{1}}, where the slope (m), is found by inserting the x -coordinate and y -coordinate from two points on a given line.

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