Graphs (Grade 9) — Mathematics Grades 8–11

Reading the gradient and intercepts of a straight-line graph, and finding its equation.

Gr 9 Term 3 — GraphsTerm 3

Practise Graphs in the app →

What gets asked

Find the x-intercept and y-intercept of a straight line.
Calculate the gradient of a straight line from two points.
Determine the equation of a line from its graph or a table.
Say whether a graph is increasing, decreasing or constant.

You must be able to

Read the y-intercept straight off an equation of the form y = mx + c.
Calculate a gradient as change in y divided by change in x.
Write the equation of a line once you know its gradient and y-intercept.
Describe the trend and shape of a graph in words.

Traps that cost marks

Calculating change in x over change in y, which flips the gradient. Gradient is change in y divided by change in x, always in that order.
Writing the equation with the gradient and intercept swapped round. Keep the standard order $y = m x + c$ : gradient first, then the y-intercept.
Setting y = 0 to find the y-intercept. Set x = 0 to find the y-intercept, and y = 0 to find the x-intercept.

Worked example

A straight line passes through (0 ; 4) and (2 ; 10). Determine its equation.

The y-intercept is where x = 0, so $c = 4$ .
Gradient $= \frac{10 - 4}{2 - 0} = 3$ .
The equation is $y = 3 x + 4$ .

Global graphs

Reading the story a graph tells, before any numbers are written on the axes.

global graph: A graph that shows the shape of a situation, without exact numbers on the axes.
linear: Making one straight line the whole way. A graph that bends is non-linear.
maximum: The highest value the graph reaches.
minimum: The lowest value the graph reaches.
discrete: Made of separate values you can count, with nothing possible in between.
continuous: Taking every value in between too, so the graph is one unbroken line.

A global graph tells a story. Its shape carries the meaning, not exact numbers. Read it from left to right, like a sentence.

Linear or not. A linear graph is one straight line from end to end. A graph built from several straight pieces at different slopes is not linear, even though each piece is straight.

Constant, increasing or decreasing. Flat means constant. Rising means increasing, and falling means decreasing. A graph that rises more slowly than before is still increasing.

Maximum and minimum. The maximum is the highest value the graph reaches, and the minimum is the lowest. The question wants that value, read off the upright axis, not the moment it happened.

Discrete or continuous. Counting learners gives discrete values, so plot separate points and leave them unjoined. Measuring water in a tank gives continuous values, so draw one unbroken line.

Rules to remember

A graph that climbs more slowly is still climbing

Examples

A tank fills at a steady rate for

10

minutes. Then the tap is closed for

5

minutes. Describe the graph.

Worked answer

Steady filling means the graph rises in a straight line.
Closing the tap means the amount of water stops changing.
So the second part is flat, and the graph is constant there.
The water can take any value in between, so the graph is continuous.

Answer: a straight rise, then a flat line

Closing the tap stopped the water rising. It did not make the water fall.

A spaza shop's bread sales rise every day for a week, but by less each day. Is the graph increasing or decreasing?

Worked answer

Sales rise every day, so each value beats the one before.
Rising by less each day means the graph flattens out.
Flattening out is not the same as falling.
So the graph is increasing, and it is not linear.

Answer: increasing, and not linear

The slope got gentler every day, but it never turned downwards.

A learner counts taxis passing the school each hour. The counts are

12

19

7

and

15

. Give the maximum, and say whether to join the points.

Worked answer

The maximum is the biggest count, which is $19$ .
It happened in the second hour, but the maximum is the count.
The minimum is $7$ , in the third hour.
Taxis are counted whole, so this data is discrete.
Plot four separate points, and do not join them up.

Answer: the maximum is $19$ , and the points stay separate

Joining the dots would claim a count existed between the hours, and none did.

Traps

Reading a graph that rises more slowly as a falling graph. Ask whether the value is still going up. If it is, the graph is increasing.
Calling a graph linear because it is built from straight pieces. A linear graph is one straight line from end to end, with no change of slope.
Giving the time at the top when the maximum value was asked for. Read the maximum off the upright axis, not off the axis along the bottom.
Drawing an unbroken curve for data that can only be whole numbers. Counted data is discrete, so plot separate points and leave the gaps in.

Learn and practise “Global graphs” in the app →

Straight-line graphs

Everything a straight line will tell you: where it crosses, how steep it is, and its equation.

gradient: How steep a line is. It is how far the line rises for each step you take across.
y-intercept: The value of $y$ where the line crosses the upright $y$ -axis.
x-intercept: The value of $x$ where the line crosses the flat $x$ -axis.

Every straight line, also called a linear graph, can be written as $y = m x + c$ . The $m$ is the gradient, and the $c$ is the $y$ -intercept. Keep them in that order. Swapping them gives a different line altogether.

Gradient is the change in $y$ divided by the change in $x$ , always that way round. A line rising to the right has a positive gradient. A line falling to the right has a negative one, and a flat line has gradient $0$ .

To find the $y$ -intercept, set $x = 0$ , because every point on the upright axis has $x$ equal to zero. To find the $x$ -intercept, set $y = 0$ . Each intercept comes from making the other letter zero.

To draw a line from its equation, work out three points in a table. Choose easy $x$ -values. Plot each point with the $x$ -value across first, then rule one straight line through all three.

$y = 4$ is a horizontal line, because $y$ stays $4$ whatever $x$ does. $x = 4$ is a vertical line. Two lines with the same gradient are parallel, so they never meet, however different the equations look.

Rules to remember

$y = m x + c$
Set $x = 0$ for the $y$ -intercept, and $y = 0$ for the $x$ -intercept

Examples

Find both intercepts of

y = 2 x - 6

Worked answer

For the $y$ -intercept set $x = 0$ : $y = 2 (0) - 6 = - 6$ .
For the $x$ -intercept set $y = 0$ : $2 x - 6 = 0$ .
Solve that: $2 x = 6$ , so $x = 3$ .
The line cuts the upright axis at $- 6$ and the flat axis at $3$ .

Answer: $y$ -intercept $- 6$ , $x$ -intercept $3$

Each intercept came from setting the other letter to $0$ , not the same one.

A straight line passes through

(0; 4)

and

(2; 10)

. Determine its equation.

Worked answer

The first point has $x = 0$ , so the $y$ -intercept is $4$ .
Change in $y$ : $10 - 4 = 6$ .
Change in $x$ : $2 - 0 = 2$ .
Gradient: $6 \div 2 = 3$ .
So the equation is $y = 3 x + 4$ .

Answer: $y = 3 x + 4$

The gradient went in front of $x$ and the intercept on its own, as the form needs.

Draw

y = - x + 3

using a table of points.

Worked answer

Choose three easy $x$ -values: $- 1$ , $0$ and $2$ .
Work out $y$ for each of them: $4$ , $3$ and $1$ .
Plot $(- 1; 4)$ , $(0; 3)$ and $(2; 1)$ , counting across first.
Rule one straight line right through all three points.
The gradient is $- 1$ , so the line falls to the right.

Answer: a straight line through $(0; 3)$ , falling to the right

Three points show up a plotting mistake, and two points never can.

Traps

Setting y = 0 to find the y-intercept. Set x = 0 to find the y-intercept, and y = 0 to find the x-intercept.
Calculating change in x over change in y, which flips the gradient. Gradient is change in y divided by change in x, always in that order.
Writing the equation with the gradient and intercept swapped round. Keep the standard order $y = m x + c$ : gradient first, then the y-intercept.
Plotting the point $(3; 5)$ as $3$ up and $5$ across. The $x$ -value is counted across first, so $(3; 5)$ is $3$ across and $5$ up.

Learn and practise “Straight-line graphs” in the app →

Grade 9 Graphs

What gets asked

You must be able to

Traps that cost marks

Worked example

Global graphs

Rules to remember

Examples

Traps

Straight-line graphs

Rules to remember

Examples

Traps

About this material