<< Chapter < Page Chapter >> Page >

Rewriting quadratics in standard form

In [link] , the quadratic was easily solved by factoring. However, there are many quadratics that cannot be factored. We can solve these quadratics by first rewriting them in standard form.

Given a quadratic function, find the x - intercepts by rewriting in standard form .

  1. Substitute a and b into h = b 2 a .
  2. Substitute x = h into the general form of the quadratic function to find k .
  3. Rewrite the quadratic in standard form using h and k .
  4. Solve for when the output of the function will be zero to find the x - intercepts.

Finding the x - Intercepts of a parabola

Find the x - intercepts of the quadratic function f ( x ) = 2 x 2 + 4 x 4.

We begin by solving for when the output will be zero.

0 = 2 x 2 + 4 x 4

Because the quadratic is not easily factorable in this case, we solve for the intercepts by first rewriting the quadratic in standard form.

f ( x ) = a ( x h ) 2 + k

We know that a = 2. Then we solve for h and k .

h = b 2 a k = f ( 1 )    = 4 2 ( 2 )    = 2 ( 1 ) 2 + 4 ( 1 ) 4    = −1    = −6

So now we can rewrite in standard form.

f ( x ) = 2 ( x + 1 ) 2 6

We can now solve for when the output will be zero.

0 = 2 ( x + 1 ) 2 6 6 = 2 ( x + 1 ) 2 3 = ( x + 1 ) 2 x + 1 = ± 3 x = 1 ± 3

The graph has x - intercepts at ( −1 3 , 0 ) and ( −1 + 3 , 0 ) .

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

In a separate Try It , we found the standard and general form for the function g ( x ) = 13 + x 2 6 x . Now find the y - and x - intercepts (if any).

y -intercept at (0, 13), No x - intercepts

Got questions? Get instant answers now!

Solving a quadratic equation with the quadratic formula

Solve x 2 + x + 2 = 0.

Let’s begin by writing the quadratic formula: x = b ± b 2 4 a c 2 a .

When applying the quadratic formula , we identify the coefficients a ,   b  and  c . For the equation x 2 + x + 2 = 0 , we have a = 1 ,   b = 1 ,   and   c = 2. Substituting these values into the formula we have:

x = b ± b 2 4 a c 2 a    = 1 ± 1 2 4 1 ( 2 ) 2 1    = 1 ± 1 8 2    = 1 ± 7 2    = 1 ± i 7 2

The solutions to the equation are x = 1 + i 7 2 and x = 1 i 7 2 or x = 1 2 + i 7 2 and x = 1 2 i 7 2 .

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

Applying the vertex and x -intercepts of a parabola

A ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. The ball’s height above ground can be modeled by the equation H ( t ) = 16 t 2 + 80 t + 40.

  1. When does the ball reach the maximum height?
  2. What is the maximum height of the ball?
  3. When does the ball hit the ground?
  1. The ball reaches the maximum height at the vertex of the parabola.
    h = 80 2 ( 16 )    = 80 32    = 5 2    = 2.5

    The ball reaches a maximum height after 2.5 seconds.

  2. To find the maximum height, find the y - coordinate of the vertex of the parabola.
    k = H ( b 2 a )    = H ( 2.5 )    = −16 ( 2.5 ) 2 + 80 ( 2.5 ) + 40    = 140

    The ball reaches a maximum height of 140 feet.

  3. To find when the ball hits the ground, we need to determine when the height is zero, H ( t ) = 0.

    We use the quadratic formula.

    t = 80 ± 80 2 4 ( 16 ) ( 40 ) 2 ( 16 )   = 80 ± 8960 32

    Because the square root does not simplify nicely, we can use a calculator to approximate the values of the solutions.

    t = 80 8960 32 5.458 or t = 80 + 8960 32 0.458

    The second answer is outside the reasonable domain of our model, so we conclude the ball will hit the ground after about 5.458 seconds. See [link]

    Graph of a negative parabola where x goes from -1 to 6.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

A rock is thrown upward from the top of a 112-foot high cliff overlooking the ocean at a speed of 96 feet per second. The rock’s height above ocean can be modeled by the equation H ( t ) = 16 t 2 + 96 t + 112.

  1. When does the rock reach the maximum height?
  2. What is the maximum height of the rock?
  3. When does the rock hit the ocean?

  1. 3 seconds
  2. 256 feet
  3. 7 seconds

Got questions? Get instant answers now!

Questions & Answers

how fast can i understand functions without much difficulty
Joe Reply
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
Practice Key Terms 6

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