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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 quadratic functions.

Functions of the form y = a ( x + p ) 2 + q

This form of the quadratic function is slightly more complex than the form studied in Grade 10, y = a x 2 + q . The general shape and position of the graph of the function of the form f ( x ) = a ( x + p ) 2 + q is shown in [link] .

Graph of f ( x ) = 1 2 ( x + 2 ) 2 - 1

Investigation : functions of the form y = a ( x + p ) 2 + q

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

From your graphs, you should have found that a affects whether the graph makes a smile or a frown. If a < 0 , the graph makes a frown and if a > 0 then the graph makes a smile. This was shown in Grade 10.

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

You should have also found that the value of p affects whether the turning point is to the left of the y -axis ( p > 0 ) or to the right of the y -axis ( p < 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 ) 2 + 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

Phet simulation for graphing

Domain and range

For f ( x ) = a ( x + p ) 2 + 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 + p ) 2 + q depends on whether the value for a is positive or negative. We will consider these two cases separately.

If a > 0 then we have:

( x + p ) 2 0 ( The square of an expression is always positive ) a ( x + p ) 2 0 ( Multiplication by a positive number maintains the nature of the inequality ) a ( x + p ) 2 + q q f ( x ) q

This tells us that for all values of x , f ( x ) is always greater than or equal to q . Therefore if a > 0 , the range of f ( x ) = a ( x + p ) 2 + q is { f ( x ) : f ( x ) [ q , ) } .

Similarly, it can be shown that if a < 0 that the range of f ( x ) = a ( x + p ) 2 + q is { f ( x ) : f ( x ) ( - , q ] } . This is left as an exercise.

For example, the domain of g ( x ) = ( x - 1 ) 2 + 2 is { x : x R } because there is no value of x R for which g ( x ) is undefined. The range of g ( x ) can be calculated as follows:

( x - p ) 2 0 ( x + p ) 2 + 2 2 g ( x ) 2

Therefore the range is { g ( x ) : g ( x ) [ 2 , ) } .

Domain and range

  1. Given the function f ( x ) = ( x - 4 ) 2 - 1 . Give the range of f ( x ) .
  2. What is the domain of the equation y = 2 x 2 - 5 x - 18 ?

Intercepts

For functions of the form, y = a ( x + p ) 2 + q , the details of calculating the intercepts with the x and y axes is given.

The y -intercept is calculated as follows:

y = a ( x + p ) 2 + q y i n t = a ( 0 + p ) 2 + q = a p 2 + q

If p = 0 , then y i n t = q .

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

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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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