Unit 5 Section 2 : Straight Line Graphs

We look in this section at how to calculate coordinates and plot straight line graphs.
We also look at the gradient and intercept of a straight line and the equation of a straight line.

The gradient of a line is a measure of its steepness. The intercept of a line is the value where the line crosses the y-axis.

The equation of a straight line is y = mx + c, where
m = gradient and c = intercept (where the line crosses the y-axis).

Example 1

Draw the graph with equation y = 2x + 3.

First, find the coordinates of some points on the graph. This can be done by calculating y for a range of x values as shown in the table.
x–2–10123
y–113579
The points can then be plotted on a set of axes and a straight line drawn through them.

Example 2

Calculate the gradient of each of the following lines:

(a)
Gradient = = 1
(b)
Gradient = = 2
(c)
Gradient = =
(d)
Gradient = = –3

Example 3

Determine the equation of each of the following lines:

(a)(b)
(a)
Gradient = = 1
Intercept = 2
So m = 1 and c = 2.
The equation is:
y = mx + c
y = 1x + 2
or
y = x + 2
(b)
Gradient = = 2
Intercept = –1
So m = 2 and c = –1.
The equation is:
y = mx + c
y = 2x + (–1)
or
y = 2x – 1

Exercises

NOTE: In these questions you may need to plot points and draw lines on graphs.
To plot a point, just click on the graph with your left mouse button. To draw a line, hold down the left mouse button at one point on the line and drag the pointer to another point on the line.
When you let go of the button the line will appear and it will automatically cross the whole graph.
If you make a mistake, press the delete key and the graph will be cleared.
Question 1
(a)
Copy and complete the following table for y = 2x – 2.
x–2–10135
y
(b)
Draw the graph of y = 2x – 2.
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Question 2
Draw the graphs with the equations given below, using a new set of axes for each graph.
(a) y = x + 3
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(b) y = x – 4
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(c) y = 4x – 1
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(d) y = 3x + 1
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(e) y = 4 – x
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(f) y = 8 – 2x
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Question 3
Calculate the gradient of each of the following lines:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Question 4
Write down the equations of the lines with gradients and intercepts listed below:
(a) Gradient = 4 and intercept = 2. y =
(b) Gradient = 2 and intercept = –5. y =
(c)
Gradient = and intercept = 1.
y =
(d) Gradient = –1 and intercept = –5. y =
Question 5
Complete the following table, which gives the equation, gradient and intercept for a number of straight lines.
Equation Gradient Intercept
y = 5x + 7
3 –2
y = –3x + 2
y = –4x – 2
– 2 3
1
y = 4 – x
y = 10 – 3x
Question 6
(a)
Plot the points A, B and C with coordinates:
A (2, 4)
B (7, 5)
C (0, 10)
and join them to form a triangle.
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(b)
Calculate the gradient of each side of the triangle.
AB:
BC:
AC:
Question 7
Determine the equation of each of the following lines:
Note: Use slash (/) to write fractions. e.g.
(a)(b)
y = y =
(c)(d)
y = y =
(e)(f)
y = y =
Question 8
(a)
On a set of axes, plot the points with coordinates
(–2, –2), (2, 0), (4, 1) and (6, 2)
and then draw a straight line through these points.
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(b)
Determine the equation of the line.
Question 9
(a)
On the same axes, draw the lines with equations y = 2x + 3 and y = 8 – x.
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(b)
Write down the coordinates of the point where the lines cross.
Question 10
The point A has coordinates (4, 2), the point B has coordinates (8, 6) and the point C has coordinates (5, 9).
(a)
Plot these points on a set of axes and draw straight lines through each point to form a triangle.
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(b)
Determine the equation of each of the lines you have drawn.
AB:
BC:
AC:
Question 11
Look at this diagram:
(a)
The line through points A and F has the equation y = 11.
What is the equation of the line through points A and B ?
(b)
The line through points A and D has the equation y = x + 3.
What is the equation of the line through points F and E ?
(c)
What is the equation of the line through points B and C ?
Question 12
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The s give the graph p = 3s + 1.
The s give the graph p = 2s + 1.
The s give the graph p = s + 1.
Selma has 16 pins.
(a)
Use the correct graph to find the number of squares she can pin up with 4 pins in each square.
How many squares can she pin up with 3 pins in each square?
squares with 4 pins in each square
squares with 3 pins in each square
(b)
The line through the points for p = 3s + 1 climbs more steeply than the line through the points for p = 2s + 1 and p = s + 1.
Which part of the equation p = 3s + 1 tells you how steep the line is?
(c)
On the grid at the beginning of this question, plot three points to show the graph for 8 pins in each square.
(d)
What is the equation of this graph?