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Introduction

In Grade 10, you studied graphs of many different forms. In this chapter, you will learn a little more about the graphs of functions.

Functions of the form y = a x + p + q

This form of the hyperbolic function is slightly more complex than the form studied in Grade 10.

General shape and position of the graph of a function of the form f ( x ) = a x + p + q . The asymptotes are shown as dashed lines.

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

  1. On the same set of axes, plot the following graphs:
    1. a ( x ) = - 2 x + 1 + 1
    2. b ( x ) = - 1 x + 1 + 1
    3. c ( x ) = 0 x + 1 + 1
    4. d ( x ) = 1 x + 1 + 1
    5. e ( x ) = 2 x + 1 + 1
    Use your results to deduce the effect of a .
  2. On the same set of axes, plot the following graphs:
    1. f ( x ) = 1 x - 2 + 1
    2. g ( x ) = 1 x - 1 + 1
    3. h ( x ) = 1 x + 0 + 1
    4. j ( x ) = 1 x + 1 + 1
    5. k ( x ) = 1 x + 2 + 1
    Use your results to deduce the effect of p .
  3. Following the general method of the above activities, choose your own values of a and p to plot 5 different graphs of y = a x + p + q to deduce the effect of q .

You should have found that the sign of a affects whether the graph is located in the first and third quadrants, or the second and fourth quadrants of Cartesian plane.

You should have also found that the value of p affects whether the x -intercept is negative ( p > 0 ) or positive ( p < 0 ).

You should have also found that the value of q affects whether the graph lies above the x -axis ( q > 0 ) or below the x -axis ( q < 0 ).

These different properties are summarised in [link] . The axes of symmetry for each graph is shown as a dashed line.

Table summarising general shapes and positions of functions of the form y = a x + p + q . The axes of symmetry are shown as dashed lines.
p < 0 p > 0
a > 0 a < 0 a > 0 a < 0
q > 0
q < 0

Domain and range

For y = a x + p + q , the function is undefined for x = - p . The domain is therefore { x : x R , x - p } .

We see that y = a x + p + q can be re-written as:

y = a x + p + q y - q = a x + p If x - p then : ( y - q ) ( x + p ) = a x + p = a y - q

This shows that the function is undefined at y = q . Therefore the range of f ( x ) = a x + p + q is { f ( x ) : f ( x ) R , f ( x ) q .

For example, the domain of g ( x ) = 2 x + 1 + 2 is { x : x R , x - 1 } because g ( x ) is undefined at x = - 1 .

y = 2 x + 1 + 2 ( y - 2 ) = 2 x + 1 ( y - 2 ) ( x + 1 ) = 2 ( x + 1 ) = 2 y - 2

We see that g ( x ) is undefined at y = 2 . Therefore the range is { g ( x ) : g ( x ) ( - , 2 ) ( 2 , ) } .

Domain and range

  1. Determine the range of y = 1 x + 1 .
  2. Given: f ( x ) = 8 x - 8 + 4 . Write down the domain of f .
  3. Determine the domain of y = - 8 x + 1 + 3

Intercepts

For functions of the form, y = a x + p + q , the intercepts with the x and y axis are calculated by setting x = 0 for the y -intercept and by setting y = 0 for the x -intercept.

The y -intercept is calculated as follows:

y = a x + p + q y i n t = a 0 + p + q = a p + q

For example, the y -intercept of g ( x ) = 2 x + 1 + 2 is given by setting x = 0 to get:

y = 2 x + 1 + 2 y i n t = 2 0 + 1 + 2 = 2 1 + 2 = 2 + 2 = 4

The x -intercepts are calculated by setting y = 0 as follows:

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

For example, the x -intercept of g ( x ) = 2 x + 1 + 2 is given by setting x = 0 to get:

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

Intercepts

  1. Given: h ( x ) = 1 x + 4 - 2 . Determine the coordinates of the intercepts of h with the x- and y-axes.
  2. Determine the x-intercept of the graph of y = 5 x + 2 . Give the reason why there is no y-intercept for this function.

Asymptotes

There are two asymptotes for functions of the form y = a x + p + q . They are determined by examining the domain and range.

We saw that the function was undefined at x = - p and for y = q . Therefore the asymptotes are x = - p and y = q .

For example, the domain of g ( x ) = 2 x + 1 + 2 is { x : x R , x - 1 } because g ( x ) is undefined at x = - 1 . We also see that g ( x ) is undefined at y = 2 . Therefore the range is { g ( x ) : g ( x ) ( - , 2 ) ( 2 , ) } .

From this we deduce that the asymptotes are at x = - 1 and y = 2 .

Asymptotes

  1. Given: h ( x ) = 1 x + 4 - 2 .Determine the equations of the asymptotes of h .
  2. Write down the equation of the vertical asymptote of the graph y = 1 x - 1 .

Sketching graphs of the form f ( x ) = a x + p + q

In order to sketch graphs of functions of the form, f ( x ) = a x + p + q , we need to calculate four characteristics:

  1. domain and range
  2. asymptotes
  3. y -intercept
  4. x -intercept

For example, sketch the graph of g ( x ) = 2 x + 1 + 2 . Mark the intercepts and asymptotes.

We have determined the domain to be { x : x R , x - 1 } and the range to be { g ( x ) : g ( x ) ( - , 2 ) ( 2 , ) } . Therefore the asymptotes are at x = - 1 and y = 2 .

The y -intercept is y i n t = 4 and the x -intercept is x i n t = - 2 .

Graph of g ( x ) = 2 x + 1 + 2 .

Graphs

  1. Draw the graph of y = 1 x + 2 . Indicate the horizontal asymptote.
  2. Given: h ( x ) = 1 x + 4 - 2 . Sketch the graph of h showing clearly the asymptotes and ALL intercepts with the axes.
  3. Draw the graph of y = 1 x and y = - 8 x + 1 + 3 on the same system of axes.
  4. Draw the graph of y = 5 x - 2 , 5 + 2 . Explain your method.
  5. Draw the graph of the function defined by y = 8 x - 8 + 4 . Indicate the asymptotes and intercepts with the axes.

End of chapter exercises

  1. Plot the graph of the hyperbola defined by y = 2 x for - 4 x 4 . Suppose the hyperbola is shifted 3 units to the right and 1 unit down. What is the new equation then ?
  2. Based on the graph of y = 1 x , determine the equation of the graph with asymptotes y = 2 and x = 1 and passing through the point (2; 3).

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Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
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