# 3.2 Quadratic functions  (Page 6/14)

 Page 6 / 14

## 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 $\text{\hspace{0.17em}}x\text{-}$ intercepts by rewriting in standard form .

1. Substitute $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ into $\text{\hspace{0.17em}}h=-\frac{b}{2a}.$
2. Substitute $\text{\hspace{0.17em}}x=h\text{\hspace{0.17em}}$ into the general form of the quadratic function to find $\text{\hspace{0.17em}}k.$
3. Rewrite the quadratic in standard form using $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}k.$
4. Solve for when the output of the function will be zero to find the $\text{\hspace{0.17em}}x\text{-}$ intercepts.

## Finding the $\text{\hspace{0.17em}}x\text{-}$ Intercepts of a parabola

Find the $\text{\hspace{0.17em}}x\text{-}$ intercepts of the quadratic function $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{2}+4x-4.$

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

$0=2{x}^{2}+4x-4$

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

$f\left(x\right)=a{\left(x-h\right)}^{2}+k$

We know that $\text{\hspace{0.17em}}a=2.\text{\hspace{0.17em}}$ Then we solve for $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}k.$

So now we can rewrite in standard form.

$f\left(x\right)=2{\left(x+1\right)}^{2}-6$

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

$\begin{array}{l}0=2{\left(x+1\right)}^{2}-6\hfill \\ 6=2{\left(x+1\right)}^{2}\hfill \\ 3={\left(x+1\right)}^{2}\hfill \\ x+1=±\sqrt{3}\hfill \\ x=-1±\sqrt{3}\hfill \end{array}$

The graph has $\text{\hspace{0.17em}}x\text{-}$ intercepts at $\text{\hspace{0.17em}}\left(-1-\sqrt{3},0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-1+\sqrt{3},0\right).$

In a separate Try It , we found the standard and general form for the function $\text{\hspace{0.17em}}g\left(x\right)=13+{x}^{2}-6x.\text{\hspace{0.17em}}$ Now find the y - and $\text{\hspace{0.17em}}x\text{-}$ intercepts (if any).

y -intercept at (0, 13), No $\text{\hspace{0.17em}}x\text{-}$ intercepts

Solve $\text{\hspace{0.17em}}{x}^{2}+x+2=0.$

Let’s begin by writing the quadratic formula: $\text{\hspace{0.17em}}x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}.$

When applying the quadratic formula , we identify the coefficients For the equation $\text{\hspace{0.17em}}{x}^{2}+x+2=0,\text{\hspace{0.17em}}$ we have Substituting these values into the formula we have:

The solutions to the equation are $\text{\hspace{0.17em}}x=\frac{-1+i\sqrt{7}}{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=\frac{-1-i\sqrt{7}}{2}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}x=\frac{-1}{2}+\frac{i\sqrt{7}}{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=\frac{-1}{2}-\frac{i\sqrt{7}}{2}.$

## 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 $\text{\hspace{0.17em}}H\left(t\right)=-16{t}^{2}+80t+40.$

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.

The ball reaches a maximum height after 2.5 seconds.

2. To find the maximum height, find the $\text{\hspace{0.17em}}y\text{-}$ coordinate of the vertex of the parabola.

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, $\text{\hspace{0.17em}}H\left(t\right)=0.$

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

$\begin{array}{l}\hfill \\ \hfill \\ \begin{array}{lll}t=\frac{-80-\sqrt{8960}}{-32}\approx 5.458\hfill & \text{or}\hfill & t=\frac{-80+\sqrt{8960}}{-32}\approx -0.458\hfill \end{array}\hfill \end{array}$

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]

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 $\text{\hspace{0.17em}}H\left(t\right)=-16{t}^{2}+96t+112.$

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

So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?