Linear Functions and Graphs

Explore Warm-up: plot the points Points: notes Fig. 6.1 · Answer box: think! Mathematics 1A, p144

Plot A(3, 2), B(−4, 1), C(−3, −2) and D(1, −3).

1. With Point switched on, click (or tap) the grid. Plot the points in order — A first, then B, C and D — so each point gets its letter.
2. Drag a point to move it. Undo removes the last point; Clear removes them all.
3. Press Check.

These are the four points of Fig. 6.1 in the notes. Remember: the x-coordinate (across) comes first, then the y-coordinate (up or down).

Investigation Equation of straight line think! Mathematics 1A, p153

Let us find out how the graph of a straight line in the form y = mx + c changes when either m or c varies.

The sliders below do the job of the textbook’s spreadsheet ‘Equation of a straight line’. The red dot marks where the line cuts the y-axis.

Worksheet Investigation 1: Different m · Investigation 2: Different c Worksheet Concept 6D · think! Mathematics 1A, p153–154

The general equation of a linear function is y = mx + c. What do m and c represent?

Each row of a list draws one line on the grid below it, labelled with its equation — the dot beside the row is the line’s colour. Edit a row, or press + Add a line to type your own — on the Investigation 1 grid (Different m), try a line with a negative gradient such as y = −2x.

Worksheet Investigation 3: The 4 cases of gradient Worksheet Concept 6D · think! Mathematics 1A, p154

Each grid shows one of the worksheet’s four lines. Follow each line with your finger from left to right, then fill in the Summary.

Summary:

Class Discussion Gradient of straight line think! Mathematics 1A, p155–156

1. Both graphs in Fig. 6.12 below show the graph of y = 2x. Why do they look different?

Fig. 6.12

2. Which of the two lines is steeper or has the larger gradient? Explain.
3. What does a gradient of 2 mean in both graphs, i.e. how do you find the gradient of the line in both graphs?

Explore Gradient explorer think! Mathematics 1A: key box p156 · Problem-solving Tip p158 · Summary p173

The textbook defines gradient like this (p156):

The dashed step always goes from the left-hand point to the right-hand point, as the textbook does it:

Using this method, the horizontal change will always be positive as it goes from left to right.

Problem-solving Tip, think! Mathematics 1A, p158.

Try it on the grid below — these steps are Math Lab’s, not the textbook’s.

1. Drag A and B. The dashed step goes from the left-hand point to the right-hand point: across first (the run), then up or down (the rise).
2. Make the line slope downwards from left to right. What happens to the rise, and to the gradient?
3. Drag B to the left of A. The step now starts at the left-hand point, B. Does the gradient change?

A horizontal or a vertical line? Its gradient is the Investigation just below this one.

Investigation Gradient of horizontal line · Gradient of vertical line think! Mathematics 1A, p160–161

Type your answers in the boxes, then press Check. Go along each segment from its first letter to its second — A to C is from left to right, as in the gradient explorer — and keep the sign: a drop is a negative rise.

Worksheet Practice 3 Worksheet Concept 6D

On a sheet of graph paper, using a scale of 1 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y-axis, draw the graph of the function y = 2x + 1 for values of x from 0 to 4.