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Functions of the form y = a x + q

Functions with a general form of y = a x + q are called straight line functions. In the equation, y = a x + q , a and q 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 [link] for the function f ( x ) = 2 x + 3 .

Graph of f ( x ) = 2 x + 3

Investigation : functions of the form y = a x + q

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

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

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

These different properties are summarised in [link] .

Table summarising general shapes and positions of graphs of functions of the form y = a x + q .
a > 0 a < 0
q > 0
q < 0

Domain and range

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

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

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

Intercepts

For functions of the form, y = a x + q , the details of calculating the intercepts with the x and y axis are given.

The y -intercept is calculated as follows:

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

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

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

The x -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

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

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

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 ) = a x + q

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

  1. sign of a
  2. y -intercept
  3. x -intercept

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

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

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

The y -intercept is obtained by setting x = 0 and was calculated earlier to be y i n t = - 1 . The x -intercept is obtained by setting y = 0 and was calculated earlier to be x i n t = 1 .

Graph of the function g ( x ) = x - 1

Draw the graph of y = 2 x + 2

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

    y = 2 ( 0 ) + 2 = 2
  2. For the intercept on the x-axis, let y = 0

    0 = 2 x + 2 2 x = - 2 x = - 1
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Exercise: straight line graphs

  1. List the y -intercepts for the following straight-line graphs:
    1. y = x
    2. y = x - 1
    3. y = 2 x - 1
    4. y + 1 = 2 x
  2. Give the equation of the illustrated graph below:
  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 + 2 y - 5 = 0 and 3 x - y - 1 = 0

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Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
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