<< Chapter < Page Chapter >> Page >

Functions of the form y = a x + q

Functions of the form y = a x + q are known as hyperbolic functions. The general form of the graph of this function is shown in [link] .

General shape and position of the graph of a function of the form f ( x ) = a x + q .

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

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

You should have found that the value of a affects whether the graph is located in the first and third quadrants of Cartesian plane.

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 are shown as a dashed line.

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

Domain and range

For y = a x + q , the function is undefined for x = 0 . The domain is therefore { x : x R , x 0 } .

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

y = a x + q y - q = a x If x 0 then : ( y - q ) ( x ) = a x = a y - q

This shows that the function is undefined at y = q . Therefore the range of f ( x ) = a x + q is { f ( x ) : f ( x ) ( - ; q ) ( q ; ) } .

For example, the domain of g ( x ) = 2 x + 2 is { x : x R , x 0 } because g ( x ) is undefined at x = 0 .

y = 2 x + 2 ( y - 2 ) = 2 x If x 0 then : x ( y - 2 ) = 2 x = 2 y - 2

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

Intercepts

For functions of the form, y = a x + q , the intercepts with the x and y axis is 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 + q y i n t = a 0 + q

which is undefined because we are dividing by 0. Therefore there is no y -intercept.

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

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

which is undefined.

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

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

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

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

Asymptotes

There are two asymptotes for functions of the form y = a x + q . Just a reminder, an asymptote is a straight or curved line, which the graph of a function will approach, but never touch. They are determined by examining the domain and range.

We saw that the function was undefined at x = 0 and for y = q . Therefore the asymptotes are x = 0 and y = q .

For example, the domain of g ( x ) = 2 x + 2 is { x : x R , x 0 } because g ( x ) is undefined at x = 0 . 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 = 0 and y = 2 .

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

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

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

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

We have determined the domain to be { x : x R , x 0 } and the range to be { g ( x ) : g ( x ) ( - ; 2 ) ( 2 ; ) } . Therefore the asymptotes are at x = 0 and y = 2 .

There is no y -intercept and the x -intercept is x i n t = - 1 .

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

Graphs

  1. Using graph (grid) paper, draw the graph of x y = - 6 .
    1. Does the point (-2; 3) lie on the graph ? Give a reason for your answer.
    2. Why is the point (-2; -3) not on the graph ?
    3. If the x -value of a point on the drawn graph is 0,25, what is the corresponding y -value ?
    4. What happens to the y -values as the x -values become very large ?
    5. With the line y = - x as line of symmetry, what is the point symmetrical to (-2; 3) ?
  2. Draw the graph of x y = 8 .
    1. How would the graph y = 8 3 + 3 compare with that of x y = 8 ? Explain your answer fully.
    2. Draw the graph of y = 8 3 + 3 on the same set of axes.

Questions & Answers

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Siyavula textbooks: grade 10 maths [ncs]. OpenStax CNX. Aug 05, 2011 Download for free at http://cnx.org/content/col11239/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: grade 10 maths [ncs]' conversation and receive update notifications?

Ask