# 3.2 Quadratic functions  (Page 6/14)

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## 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

how fast can i understand functions without much difficulty
what is set?
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
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?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
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
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.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
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?
I want to learn about the law of exponent
explain this