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

There are also y intercept worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

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

The **intercept **of a graph is where it **crosses the coordinate axes**.

You can find the y and x intercepts of graphs of all types of functions including straight lines, quadratic functions, cubic functions and others. The x intercepts of a quadratic equation are also known as its roots. In A Level Mathematics we look at finding intercepts of more complicated functions.

To find the intercepts,

Substitute x=0 into the equation of the function to find the y intercept.

Substitute y=0 into the equation of the function to find the x intercept.

Let’s look at some examples,

A straight line has the equation 4x-3y=18. Find the y and x intercepts of the line.

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

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).

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

Now solve the equation to find the x value.

\begin{aligned} 4x&=18\\\\ x&=4.5 \end{aligned}The x intercept is 4.5 and has coordinate (4.5, \ 0).

- If any function is of the form y=[ function of x] \ + constant, the constant is the y intercept.

For example,

In order to find the y intercept and the x intercept:

**Substitute**x=0**into the equation of the line.****Solve the equation to find the**\bf{y}**intercept.****Substitute**y=0**into the equation of the line.****Solve the equation to find the**\bf{x}**intercept**.

Get your free y intercept worksheet of 20+ straight line graph questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free y intercept worksheet of 20+ straight line graph questions and answers. Includes reasoning and applied questions.

COMING SOONFind the y intercept and x intercept of the line y=2x-5.

**Substitute**x=0**into the equation of the line.**

2**Solve the equation to find the ** \bf{y} ** intercept.**

This equation gives y=-5.

The y intercept is -5.

It has coordinates (0,-5).

You might have noticed that this is the special case where the equation is in the form y=[ function of x] \ + \ c. In this case c=-5 so the y intercept is (0, -5).

3**Substitute ** y=0 ** into the equation of the line.**

4**Solve the equation to find the ** \bf{x} ** intercept**.

The x intercept is 2.5.

It has coordinates (2.5, \ 0).

Find the y intercept and x intercept of the line y=\frac{1}{2}x+3.

**Substitute ** x=0 ** into the equation of the line.**

y=\frac{1}{2}(0)+3

** Solve the equation to find the** \bf{y}

This equation gives y=3.

The y intercept is 3 .

It has coordinates (0, \ 3).

**Substitute ** y=0 ** into the equation of the line.**

0=\frac{1}{2}x+3

**Solve the equation to find the ** \bf{x} ** intercept**.

\begin{aligned} 0&=\frac{1}{2}x+3\\\\ \frac{1}{2}x&=-3\\\\ x&=-6 \end{aligned}

The x intercept is -6.

It has coordinates (-6, \ 0).

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

**Substitute ** x=0 ** into the equation of the line.**

2(0)+5y=20

** Solve the equation to find the ** \bf{y}

\begin{aligned}
5y&=20\\\\
y&=4
\end{aligned}

The y intercept is 4 .

It has coordinates (0, \ 4).

**Substitute ** y=0 ** into the equation of the line.**

2x+5(0)=20

**Solve the equation to find the ** \bf{x} ** intercept**.

\begin{aligned}
2x&=20\\\\
x&=10
\end{aligned}

The x intercept is 10.

It has coordinates (10, \ 0).

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

**Substitute ** x=0 ** into the equation of the line.**

3(0)-4y=24

** Solve the equation to find the ** \bf{y}

\begin{aligned}
-4y&=24\\\\
y&=-6
\end{aligned}

The y intercept is -6 .

It has coordinates (0, \ -6).

**Substitute ** y=0 ** into the equation of the line.**

3x-4(0)=24

**Solve the equation to find the ** \bf{x} ** intercept**.

\begin{aligned}
3x&=24\\\\
x&=8
\end{aligned}

The x intercept is 8.

It has coordinates (8, \ 0).

James is measuring the height of an ice sculpture as it melts.

He thinks the height, h \ cm, can be modelled using the equation h=50-2t, where t is the number of hours after the sculpture was made.

Find the initial height of the sculpture and the time taken for it to melt completely according to James’ equation.

**Substitute ** t=0 ** into the equation of the line.**

In this problem x is represented by t and y by h.

The initial height is the height at the start of the experiment. At the start of the experiment, no time has passed so t=0.

The initial height of the sculpture is the h intercept when t = 0.

h=50-2(0)

** Solve the equation to find the ** \bf{h}

h=50

The h intercept is 50.

The initial height of the sculpture is 50 \ cm.

**Substitute ** h=0 ** into the equation of the line.**

When the sculpture has melted completely, the height of the sculpture would be 0.

The time taken for the sculpture to melt completely is the t intercept when h = 0.

0=50-2t

**Solve the equation to find the ** \bf{t} ** intercept**.

\begin{aligned}
2t&=50\\\\
t&=25
\end{aligned}

The t intercept is 25.

It takes 25 hours for the sculpture to melt completely.

**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, \ c), or (a, \ 0) for the coordinates of the x intercept. ** **

**Confusing the intercept with the gradient**

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

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

(-2,0)

(0,-2)

(3,0)

(0,3)

The coordinate of the y intercept of a line in the form y=mx+c will be (0,c).

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

(2,0)

(0,2)

(-2,0)

(-6,0)

Substitute y=0 to give

\begin{aligned} 0&=3x-6\\\\ 3x&=6 \\\\ x&=2 \end{aligned}

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

Find the y intercept and x intercept of the line.

y intercept = \frac{8}{3}, \ x intercept = 8

y intercept = 8, \ x intercept = -3

y intercept = 8, \ x intercept = -\frac{8}{3}

y intercept = 8, \ x intercept = \frac{8}{3}

When x=0, \ y=8-3(0)=8

When y=0,

\begin{aligned} 0&=8-3x\\\\ 3x&=8 \\\\ x&=\frac{8}{3} \end{aligned}

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

Find the y intercept and x intercept of the line.

y intercept = -3, \ x intercept = 9

y intercept = 9, \ x intercept = -3

y intercept = -\frac{1}{3}, \ x intercept = 9

y intercept = 9, \ x intercept = -\frac{1}{3}

When x=0,

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

When y=0,

\begin{aligned} x-3(0)&=9\\\\ x&=9 \end{aligned}

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

Find the y intercept and x intercept of the line.

y intercept = 2, \ x intercept = -2.5

y intercept = -2.5, \ x intercept = 2

y intercept = -\frac{2}{5}, \ x intercept = \frac{1}{2}

y intercept = \frac{1}{2}, \ x intercept = -\frac{2}{5}

When x=0,

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

When y=0,

\begin{aligned} 5x-4(0)&=10\\\\ 5x&=10\\\\ x&=2 \end{aligned}

6. An object is projected vertically upwards and its velocity, v, is modelled by the equation v=14-10t, where t is the time in seconds after its initial projection.

Find the time when the object reaches its maximum height.

1.4 seconds

14 seconds

10 seconds

0.71 seconds

Maximum height occurs when v = 0, which is the t intercept of the line equation.

\begin{aligned} 0&=14-10t\\\\ 10t&=14\\\\ t&=1.4 \end{aligned}

1. The cost £C of hiring a taxi for a journey of m miles is given by the formula,

C=0.6m+5.The cost is plotted on the following set of axes.

(a) State the y intercept of the line.

(b) Give an interpretation of the y intercept.

**(2 marks)**

Show answer

(a) 5

**(1)**

(b) The price of hiring the taxi before it has travelled any distance / a fixed cost added to the price of each journey.

**(1)**

2. A line has the equation 6x-3y=12.

(a) Find the y intercept of the line.

(b) Find the x intercept of the line.

**(4 marks)**

Show answer

(a)

Substituting x=0.

**(1)**

Solving to give y = – 4.

**(1)**

(b)

Substituting y=0.

**(1)**

Solving to give 2.

**(1)**

3. The line L passes through the point (3,5) and has gradient 2.

The line crosses the x axis at point A.

Write down the coordinates of point A.

**(4 marks)**

Show answer

Method using the point (3,5) and gradient of 2 to find the equation.

**(1)**

Equation of y=2x-1 found.

**(1)**

Substituting y=0.

**(1)**

Solving to give x=\frac{1}{2} \ \text{or} \ (\frac{1}{2},0).

**(1)**

You have now learned how to:

- Find the y intercept and x intercept of straight line graphs

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