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The straight line

Functions of the form y=ax+qy=ax+q

Functions with a general form of y=ax+qy=ax+q are called straight line functions. In the equation, y=ax+qy=ax+q, aa and qq are constants and have different effects on the graph of the function. The general shape of the graph of functions of this form is shown in Figure 1 for the function f(x)=2x+3f(x)=2x+3.

Investigation : Functions of the Form y=ax+qy=ax+q

1. On the same set of axes, plot the following graphs:
1. a(x)=x-2a(x)=x-2
2. b(x)=x-1b(x)=x-1
3. c(x)=xc(x)=x
4. d(x)=x+1d(x)=x+1
5. e(x)=x+2e(x)=x+2
Use your results to deduce the effect of different values of qq on the resulting graph.
2. On the same set of axes, plot the following graphs:
1. f(x)=-2·xf(x)=-2·x
2. g(x)=-1·xg(x)=-1·x
3. h(x)=0·xh(x)=0·x
4. j(x)=1·xj(x)=1·x
5. k(x)=2·xk(x)=2·x
Use your results to deduce the effect of different values of aa on the resulting graph.

You may have noticed that the value of aa affects the slope of the graph. As aa increases, the slope of the graph increases. If a>0a>0 then the graph increases from left to right (slopes upwards). If a<0a<0 then the graph increases from right to left (slopes downwards). For this reason, aa is referred to as the slope or gradient of a straight-line function.

You should have also found that the value of qq affects where the graph passes through the yy-axis. For this reason, qq is known as the y-intercept.

These different properties are summarised in Table 1.

 a > 0 a > 0 a < 0 a < 0 q > 0 q > 0 q < 0 q < 0

Domain and Range

For f(x)=ax+qf(x)=ax+q, the domain is {x:xR}{x:xR} because there is no value of xRxR for which f(x)f(x) is undefined.

The range of f(x)=ax+qf(x)=ax+q is also {f(x):f(x)R}{f(x):f(x)R} because there is no value of f(x)Rf(x)R for which f(x)f(x) is undefined.

For example, the domain of g(x)=x-1g(x)=x-1 is {x:xR}{x:xR} because there is no value of xRxR for which g(x)g(x) is undefined. The range of g(x)g(x) is {g(x):g(x)R}{g(x):g(x)R}.

Intercepts

For functions of the form, y=ax+qy=ax+q, the details of calculating the intercepts with the xx and yy axis are given.

The yy-intercept is calculated as follows:

y = a x + q y i n t = a ( 0 ) + q = q y = a x + q y i n t = a ( 0 ) + q = q
(1)

For example, the yy-intercept of g(x)=x-1g(x)=x-1 is given by setting x=0x=0 to get:

g ( x ) = x - 1 y i n t = 0 - 1 = - 1 g ( x ) = x - 1 y i n t = 0 - 1 = - 1
(2)

The xx-intercepts are calculated as follows:

y = a x + q 0 = a · x i n t + q a · x i n t = - q x i n t = - q a y = a x + q 0 = a · x i n t + q a · x i n t = - q x i n t = - q a
(3)

For example, the xx-intercepts of g(x)=x-1g(x)=x-1 is given by setting y=0y=0 to get:

g ( x ) = x - 1 0 = x i n t - 1 x i n t = 1 g ( x ) = x - 1 0 = x i n t - 1 x i n t = 1
(4)

Turning Points

The graphs of straight line functions do not have any turning points.

Axes of Symmetry

The graphs of straight-line functions do not, generally, have any axes of symmetry.

Sketching Graphs of the Form f(x)=ax+qf(x)=ax+q

In order to sketch graphs of the form, f(x)=ax+qf(x)=ax+q, we need to determine three characteristics:

1. sign of aa
2. yy-intercept
3. xx-intercept

Only two points are needed to plot a straight line graph. The easiest points to use are the xx-intercept (where the line cuts the xx-axis) and the yy-intercept.

For example, sketch the graph of g(x)=x-1g(x)=x-1. Mark the intercepts.

Firstly, we determine that a>0a>0. This means that the graph will have an upward slope.

The yy-intercept is obtained by setting x=0x=0 and was calculated earlier to be yint=-1yint=-1. The xx-intercept is obtained by setting y=0y=0 and was calculated earlier to be xint=1xint=1.

Exercise 1: Drawing a straight line graph

Draw the graph of y=2x+2y=2x+2

Solution
1. Step 1. Find the y-intercept :

To find the intercept on the y-axis, let x=0x=0

y = 2 ( 0 ) + 2 = 2 y = 2 ( 0 ) + 2 = 2
(5)
2. Step 2. Find the x-intercept :

For the intercept on the x-axis, let y=0y=0

0 = 2 x + 2 2 x = - 2 x = - 1 0 = 2 x + 2 2 x = - 2 x = - 1
(6)
3. Step 3. Draw the graph by marking the two coordinates and joining them :

Exercise: Straight line graphs

1. List the yy-intercepts for the following straight-line graphs:
1. y=xy=x
2. y=x-1y=x-1
3. y=2x-1y=2x-1
4. y+1=2xy+1=2x
2. Give the equation of the illustrated graph below: Click here for the solution
3. Sketch the following relations on the same set of axes, clearly indicating the intercepts with the axes as well as the co-ordinates of the point of interception of the graph: x+2y-5=0x+2y-5=0 and 3x-y-1=03x-y-1=0

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