Straight Line Graphs

A linear expression in x is usually written as ax + b, where a and b are constants (which can be any number). Examples are: 5x + 1, 4x – 3, 2 – 7x. 

Note:
The power of x and y must be 1 for an expression to be linear. A linear graph has two variables: The dependent variable (y) and the independent variable (x). 

Example 12.1.1: 

Draw the graph of y = 2x + 3 

Solution

Step 1: Prepare the table by:

a. Choosing various values of x, including negative values, positive values and zero. e.g. x = -3, -2, -1, 0, 1, 2, (i.e. from -3 to 2). \( \scriptsize -3 \: \leq x \: \leq 2 \) 

b. Calculate the value of y for each value of x in the expression y = 2x + 3. 

x  = -3	y = 2x + 3  y = 2(-3) + 3    y = -6 + 3 y = -3
x = -2	y = 2 (-2) + 3)    y =-4 + 3 y = -1
x  = -1	y  = 2(-1) + 3 y = -2 + 3 y = 1
x  = 0	y = 2(0) + 3 y = 3
x  = 1	y = 2 (1) + 3 y = 2 + 3 y = 5
x  = 2	y = 2 (2) + 3 y = 4 + 3 y = 7

x	-3	-2	-1	0	1	2
y	-3	-1	1	3	5	7

Step 2:  Plot the points from the table of values by using the coordinate pairs 

 (-3, -3), (-2, -1), (-1, 1), (0, 3), (1, 5),  (2, 7)

Example 12.1.2:

a. Draw the graph of 2x – y = 5 for values of x = -2 to 5  -2≤x≤5
b. Use the graph to find the value of y when x = 1.5
c. Use the graph to solve 2x – 5 = 0 
d. Write down the coordinates of the points where the graph cuts the y-axis 

Solution

When x = -2	y = 2 (-2) -5) y = -4 – 5 y = -9
When x = -1	y = 2 (-1) -5) y = -2 – 5 y = -7
When x = 0	y = 2 (0) -5) y = 0 – 5 y = -5
When x = 1	y = 2 (1) -5) y = 2 – 5 y = -3
When x = 2	y = 2 (2) -5) y = 4 – 5 y = -1
When x = 3	y = 2 (3) -5) y = 6 – 5 y = 1
When x = 4	y = 2 (4) -5) y = 8 – 5 y = 3
When x = 5	y = 2 (5) -5) y = 10 – 5 y = 5

x	-2	-1	0	1	2	3	4	5
y	-9	-7	-5	-3	-1	1	3	5

a.

b. Draw a vertical line to touch the line of the graph. Mark the point and trace it to the y-axis.

y = -2, when x = 1.5

c. The solution of 2x – 5 = 0 is the x ordinate where the graph of y = 2x – 5 crosses the x-axis 

Solution x = 2.5

d. The line crosses the y-axis at (0,-5)