1. Home
  2. Precalculus
  3. Polynomial and rational functions
  4. Quadratic functions

3.2 Quadratic functions  (Page 3/14)

<< Chapter < Page
  Precalculus     Page 3 / 14
Chapter >> Page >

Writing the equation of a quadratic function from the graph

Write an equation for the quadratic function   g   in [link] as a transformation of   f ( x ) = x 2 ,   and then expand the formula, and simplify terms to write the equation in general form.

Graph of a parabola with its vertex at (-2, -3).

We can see the graph of g is the graph of   f ( x ) = x 2   shifted to the left 2 and down 3, giving a formula in the form   g ( x ) = a ( x + 2 ) 2 – 3.

Substituting the coordinates of a point on the curve, such as   ( 0 , −1 ) ,   we can solve for the stretch factor.

− 1 = a ( 0 + 2 ) 2 − 3     2 = 4 a     a = 1 2

In standard form, the algebraic model for this graph is   ( g ) x = 1 2 ( x + 2 ) 2 – 3.

To write this in general polynomial form, we can expand the formula and simplify terms.

g ( x ) = 1 2 ( x + 2 ) 2 − 3         = 1 2 ( x + 2 ) ( x + 2 ) − 3         = 1 2 ( x 2 + 4 x + 4 ) − 3         = 1 2 x 2 + 2 x + 2 − 3         = 1 2 x 2 + 2 x − 1

Notice that the horizontal and vertical shifts of the basic graph of the quadratic function determine the location of the vertex of the parabola; the vertex is unaffected by stretches and compressions.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

A coordinate grid has been superimposed over the quadratic path of a basketball in [link] . Find an equation for the path of the ball. Does the shooter make the basket?

Stop motioned picture of a boy throwing a basketball into a hoop to show the parabolic curve it makes.
(credit: modification of work by Dan Meyer)

The path passes through the origin and has vertex at   ( − 4 ,   7 ) ,   so   ( h ) x = – 7 16 ( x + 4 ) 2 + 7.   To make the shot,   h ( − 7.5 )   would need to be about 4 but   h ( – 7.5 ) ≈ 1.64 ;   he doesn’t make it.

Got questions? Get instant answers now!

Given a quadratic function in general form, find the vertex of the parabola.

  1. Identify   a ,   b ,   and   c .
  2. Find   h ,   the x -coordinate of the vertex, by substituting   a   and   b   into   h = – b 2 a .
  3. Find   k ,   the y -coordinate of the vertex, by evaluating   k = f ( h ) = f ( − b 2 a ) .

Finding the vertex of a quadratic function

Find the vertex of the quadratic function   f ( x ) = 2 x 2 – 6 x + 7.   Rewrite the quadratic in standard form (vertex form).

The horizontal coordinate of the vertex will be at

h = – b 2 a    = – – 6 2 ( 2 )    = 6 4    = 3 2

The vertical coordinate of the vertex will be at

k = f ( h )    = f ( 3 2 )    = 2 ( 3 2 ) 2 − 6 ( 3 2 ) + 7    = 5 2

Rewriting into standard form, the stretch factor will be the same as the   a   in the original quadratic.

f ( x ) = a x 2 + b x + c f ( x ) = 2 x 2 − 6 x + 7

Using the vertex to determine the shifts,

f ( x ) = 2 ( x – 3 2 ) 2 + 5 2
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Given the equation   g ( x ) = 13 + x 2 − 6 x , write the equation in general form and then in standard form.

g ( x ) = x 2 − 6 x + 13   in general form;   g ( x ) = ( x − 3 ) 2 + 4   in standard form

Got questions? Get instant answers now!

Finding the domain and range of a quadratic function

Any number can be the input value of a quadratic function. Therefore, the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum point, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all y -values greater than or equal to the y -coordinate at the turning point or less than or equal to the y -coordinate at the turning point, depending on whether the parabola opens up or down.

Domain and range of a quadratic function

The domain of any quadratic function is all real numbers.

The range of a quadratic function written in general form   f ( x ) = a x 2 + b x + c   with a positive   a   value is   f ( x ) ≥ f ( − b 2 a ) ,   or   [ f ( − b 2 a ) , ∞ ) ;   the range of a quadratic function written in general form with a negative   a   value is   f ( x ) ≤ f ( − b 2 a ) ,   or   ( − ∞ , f ( − b 2 a ) ] .

The range of a quadratic function written in standard form   f ( x ) = a ( x − h ) 2 + k   with a positive   a   value is   f ( x ) ≥ k ;   the range of a quadratic function written in standard form with a negative   a   value is   f ( x ) ≤ k .

Questions & Answers

What are types of cell
Nansoh Reply
how can I get this book
Gatyin Reply
what is lump
Chineye Reply
what is cell
Maluak Reply
what is biology
Maluak
what's cornea?
Majak Reply
what are cell
Achol
Explain the following terms . (1) Abiotic factors in an ecosystem
Nomai Reply
Abiotic factors are non living components of ecosystem.These include physical and chemical elements like temperature,light,water,soil,air quality and oxygen etc
Qasim
what is biology
daniel Reply
what is diffusion
Emmanuel Reply
passive process of transport of low-molecular weight material according to its concentration gradient
AI-Robot
what is production?
Catherine
Pathogens and diseases
Reply
how did the oxygen help a human being
Achol Reply
how did the nutrition help the plants
Achol Reply
Biology is a branch of Natural science which deals/About living Organism.
Ahmedin Reply
what is phylogeny
Odigie Reply
evolutionary history and relationship of an organism or group of organisms
AI-Robot
ok
Deng
what is biology
Hajah Reply
cell is the smallest unit of the humanity biologically
Abraham
ok
Achol
what is biology
Victoria Reply
what is biology
Abraham
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 6

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask