Algebra

Straight-line graphs

Change steepness. Connect points and a rule.

Predict, test and explain

Step 1 of 3

Predict

Change y = 2x + 1 to y = 2x + 3. What happens to the gradient?

y=x+1y = x + 1
−10-10
−10-10
−8-8
−8-8
−6-6
−6-6
−4-4
−4-4
−2-2
−2-2
22
22
44
44
66
66
88
88
1010
1010
xx
yy
(−2,  −1)(-2,\;-1)
(1,  2)(1,\;2)

Scroll the diagram sideways to see every label.

Rise / runm=33=1m=\frac{3}{3}=1Intercept(0,  1)(0,\;1)

InvestigateKeep the gradient fixed. What stays the same as the intercept changes?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

What is the gradient of this line?

y=−3x+4y=-3x+4

Hint

The coefficient of the input gives the change in output per unit step.

Reveal answer

The output decreases by three for each unit increase in the input.

m=−3m=-3

Build an explanation

Hint 1

Find where the line crosses the vertical axis.

Hint 2

Use two points: gradient is vertical change divided by horizontal change.

Hint 3

In y = mx + c, m is the gradient and c is the y-intercept.

Worked example

This example uses fixed values, separate from the diagram controls.

For y = 2x + 1, find the intercept and y when x = 3.

  1. At x = 0, y = 1, so the intercept is (0, 1).
  2. Substitute x = 3: y = 2 × 3 + 1.

Answer: Intercept (0, 1); y = 7 when x = 3.

Connect this idea

Watch out: Gradient is rise divided by run; the intercept is a point, not the slope.