Cartesian Coordinates:
Functions & Linear Graphs
Learning Objectives:
1. State the coordinates of a point
2. Plot a point in a Cartesian plane
3. Draw the Graph of a linear function
4. Find the gradient of a straight line
5. State the y-intercept of a straight line
6. solve problems involving linear graphs in real-world contexts
A function is a relationship between two variables x and y such that every input x produces exactly one output y.
The cartesian plane consists of two number lines intersecting at right angles at point 0, known as the origin.
The position of any point in the plane can be determined by its distance from each of its axes.
(Draw the Graph of a Linear Function)
Ex. 1.
i) On a sheet of graph paper, using a scale of 2cm 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 for values of x from 0 to 4.
ii) The point (3, p) lies on the graph in (i), and find the value of p.
Solution:
We first set up a table of values for x and y. These pairs of values for x and y satisfy the equation of the function y = 2x.
x 0 2 4
y = 2x 0 4 8
Using the scale of 2 cm to represent 1 unit on the x-axis and
1 cm to represent 1 unit on the y-axis, the three pairs of values are plotted as points in the Cartesian plane and a straight line is drawn to pass through these points.
Q.2. i) On a sheet of graph paper, using a scale of 2 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.
ii) The point (q, 6) lies on the graph in (i). Find the value of q.
Q.3. On the sheet of graph paper, using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y - axis, draw the graphs of the functions y = 3x and y = 2 - 2x for values of x from - 2 to 2.
Drawing the Grap of a Linear Function:
i) On a sheet of graph paper, using a scale of 2 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 for values of x from 0 to 4.
ii) The point (3, p) lies on the graph in (i). Find the value of p.
iii) From the graph in (i)
when x= 3, p = y = 6
Practice 1.
1. i) On a sheet of graph paper, using a scale of 2 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.
ii) The point (q, 6) lies on the graph in (i). find the value of q.
2. On a sheet of graph paper, using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y-axis, draw the graphs of the function y = 3x and y = 2 - 2x for values of x from - 2 to 2.
Let O be the origin (0,0).
Vertically change (or rise) OP = 6
Horizontal change (or run) OQ = 2
Since the line slopes downwards from the left to the right, its gradient is negative.
Gradient = rise/run
= - 6/2
= -3
Distance -Time Graph:
Q.1. The travel graph shows a journey taken by a cyclist. He started his 50-km journey at 0800 hours. At 0900 hours, his bicycle tire suffered a puncture and he spent half an hour repairing it. He then continued his journey and reached his destination at 11.30 hours.
a) How far did the cyclist travel before his bicycle tire suffered a puncture?
b) Find the gradient of each of the following line segments, stating clearly what each gradient represents.
i) OA
ii) AB
iii) BC
Solution:
a) 20 km
b) i) Gradient of OA = 20/I
= 20
The cyclist traveled 20 km in 1 hour, i.e. his average speed was 20 km/h
before his bicycle tire suffered a puncture
ii) Gradient of AB = 0
He stopped cycling, i.e. his average speed was zero.
iii) Gradient of BC = 30/2
= 15
The average speed during the last part of his journey was 15 km/h.
