# 3.2 Quadratic functions  (Page 8/14)

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$f\left(x\right)=-\frac{1}{3}{x}^{2}-2x+3$

For the following exercises, determine the domain and range of the quadratic function.

$f\left(x\right)={\left(x-3\right)}^{2}+2$

Domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right).\text{\hspace{0.17em}}$ Range is $\text{\hspace{0.17em}}\left[2,\infty \right).$

$f\left(x\right)=-2{\left(x+3\right)}^{2}-6$

$f\left(x\right)={x}^{2}+6x+4$

Domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right).\text{\hspace{0.17em}}$ Range is $\text{\hspace{0.17em}}\left[-5,\infty \right).$

$f\left(x\right)=2{x}^{2}-4x+2$

$k\left(x\right)=3{x}^{2}-6x-9$

Domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right).\text{\hspace{0.17em}}$ Range is $\text{\hspace{0.17em}}\left[-12,\infty \right).$

For the following exercises, solve the equations over the complex numbers.

${x}^{2}=-25$

${x}^{2}=-8$

$\left\{2i\sqrt{2},-2i\sqrt{2}\right\}$

${x}^{2}+36=0$

${x}^{2}+27=0$

$\left\{3i\sqrt{3},-3i\sqrt{3}\right\}$

${x}^{2}+2x+5=0$

${x}^{2}-4x+5=0$

$\left\{2+i,2-i\right\}$

${x}^{2}+8x+25=0$

${x}^{2}-4x+13=0$

$\left\{2+3i,2-3i\right\}$

${x}^{2}+6x+25=0$

${x}^{2}-10x+26=0$

$\left\{5+i,5-i\right\}$

${x}^{2}-6x+10=0$

$x\left(x-4\right)=20$

$x\left(x-2\right)=10$

$2{x}^{2}+2x+5=0$

$5{x}^{2}-8x+5=0$

$5{x}^{2}+6x+2=0$

$2{x}^{2}-6x+5=0$

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

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

For the following exercises, use the vertex $\text{\hspace{0.17em}}\left(h,k\right)\text{\hspace{0.17em}}$ and a point on the graph $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ to find the general form of the equation of the quadratic function.

$\left(h,k\right)=\left(2,0\right),\left(x,y\right)=\left(4,4\right)$

$f\left(x\right)={x}^{2}-4x+4$

$\left(h,k\right)=\left(-2,-1\right),\left(x,y\right)=\left(-4,3\right)$

$\left(h,k\right)=\left(0,1\right),\left(x,y\right)=\left(2,5\right)$

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

$\left(h,k\right)=\left(2,3\right),\left(x,y\right)=\left(5,12\right)$

$\left(h,k\right)=\left(-5,3\right),\left(x,y\right)=\left(2,9\right)$

$f\left(x\right)=\frac{6}{49}{x}^{2}+\frac{60}{49}x+\frac{297}{49}$

$\left(h,k\right)=\left(3,2\right),\left(x,y\right)=\left(10,1\right)$

$\left(h,k\right)=\left(0,1\right),\left(x,y\right)=\left(1,0\right)$

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

$\left(h,k\right)=\left(1,0\right),\left(x,y\right)=\left(0,1\right)$

## Graphical

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.

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

Vertex Axis of symmetry is $\text{\hspace{0.17em}}x=1.\text{\hspace{0.17em}}$ Intercepts are

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

$f\left(x\right)={x}^{2}-5x-6$

Vertex $\text{\hspace{0.17em}}\left(\frac{5}{2},\frac{-49}{4}\right),\text{\hspace{0.17em}}$ Axis of symmetry is $\text{\hspace{0.17em}}\left(0,-6\right),\left(-1,0\right),\left(6,0\right).$

$f\left(x\right)={x}^{2}-7x+3$

$f\left(x\right)=-2{x}^{2}+5x-8$

Vertex Axis of symmetry is $\text{\hspace{0.17em}}x=\frac{5}{4}.\text{\hspace{0.17em}}$ Intercepts are

$f\left(x\right)=4{x}^{2}-12x-3$

For the following exercises, write the equation for the graphed function.

$f\left(x\right)={x}^{2}-4x+1$

$f\left(x\right)=-2{x}^{2}+8x-1$

$f\left(x\right)=\frac{1}{2}{x}^{2}-3x+\frac{7}{2}$

## Numeric

For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.

 $x$ –2 –1 0 1 2 $y$ 5 2 1 2 5

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

 $x$ –2 –1 0 1 2 $y$ 1 0 1 4 9
 $x$ –2 –1 0 1 2 $y$ –2 1 2 1 –2

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

 $x$ –2 –1 0 1 2 $y$ –8 –3 0 1 0
 $x$ –2 –1 0 1 2 $y$ 8 2 0 2 8

$f\left(x\right)=2{x}^{2}$

## Technology

For the following exercises, use a calculator to find the answer.

Graph on the same set of axes the functions

What appears to be the effect of changing the coefficient?

Graph on the same set of axes $\text{\hspace{0.17em}}f\left(x\right)={x}^{2},f\left(x\right)={x}^{2}+2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(x\right)={x}^{2},f\left(x\right)={x}^{2}+5\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}-3.\text{\hspace{0.17em}}$ What appears to be the effect of adding a constant?

The graph is shifted up or down (a vertical shift).

Graph on the same set of axes

What appears to be the effect of adding or subtracting those numbers?

The path of an object projected at a 45 degree angle with initial velocity of 80 feet per second is given by the function $\text{\hspace{0.17em}}h\left(x\right)=\frac{-32}{{\left(80\right)}^{2}}{x}^{2}+x\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is the horizontal distance traveled and $\text{\hspace{0.17em}}h\left(x\right)\text{\hspace{0.17em}}$ is the height in feet. Use the TRACE feature of your calculator to determine the height of the object when it has traveled 100 feet away horizontally.

50 feet

A suspension bridge can be modeled by the quadratic function $\text{\hspace{0.17em}}h\left(x\right)=.0001{x}^{2}\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}-2000\le x\le 2000\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}|x|\text{\hspace{0.17em}}$ is the number of feet from the center and $\text{\hspace{0.17em}}h\left(x\right)\text{\hspace{0.17em}}$ is height in feet. Use the TRACE feature of your calculator to estimate how far from the center does the bridge have a height of 100 feet.

## Extensions

For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.

Vertex $\text{\hspace{0.17em}}\left(1,-2\right),\text{\hspace{0.17em}}$ opens up.

Domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right).\text{\hspace{0.17em}}$ Range is $\text{\hspace{0.17em}}\left[-2,\infty \right).$

Vertex $\text{\hspace{0.17em}}\left(-1,2\right)\text{\hspace{0.17em}}$ opens down.

Vertex $\text{\hspace{0.17em}}\left(-5,11\right),\text{\hspace{0.17em}}$ opens down.

Domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)\text{\hspace{0.17em}}$ Range is $\text{\hspace{0.17em}}\left(-\infty ,11\right].$

Vertex $\text{\hspace{0.17em}}\left(-100,100\right),\text{\hspace{0.17em}}$ opens up.

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.

Contains $\text{\hspace{0.17em}}\left(1,1\right)\text{\hspace{0.17em}}$ and has shape of $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{2}.\text{\hspace{0.17em}}$ Vertex is on the $\text{\hspace{0.17em}}y\text{-}$ axis.

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

Contains $\text{\hspace{0.17em}}\left(-1,4\right)\text{\hspace{0.17em}}$ and has the shape of $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{2}.\text{\hspace{0.17em}}$ Vertex is on the $\text{\hspace{0.17em}}y\text{-}$ axis.

Contains $\text{\hspace{0.17em}}\left(2,3\right)\text{\hspace{0.17em}}$ and has the shape of $\text{\hspace{0.17em}}f\left(x\right)=3{x}^{2}.\text{\hspace{0.17em}}$ Vertex is on the $\text{\hspace{0.17em}}y\text{-}$ axis.

$f\left(x\right)=3{x}^{2}-9$

Contains $\text{\hspace{0.17em}}\left(1,-3\right)\text{\hspace{0.17em}}$ and has the shape of $\text{\hspace{0.17em}}f\left(x\right)=-{x}^{2}.\text{\hspace{0.17em}}$ Vertex is on the $\text{\hspace{0.17em}}y\text{-}$ axis.

Contains $\text{\hspace{0.17em}}\left(4,3\right)\text{\hspace{0.17em}}$ and has the shape of $\text{\hspace{0.17em}}f\left(x\right)=5{x}^{2}.\text{\hspace{0.17em}}$ Vertex is on the $\text{\hspace{0.17em}}y\text{-}$ axis.

$f\left(x\right)=5{x}^{2}-77$

Contains $\text{\hspace{0.17em}}\left(1,-6\right)\text{\hspace{0.17em}}$ has the shape of $\text{\hspace{0.17em}}f\left(x\right)=3{x}^{2}.\text{\hspace{0.17em}}$ Vertex has x-coordinate of $\text{\hspace{0.17em}}-1.$

## Real-world applications

Find the dimensions of the rectangular corral producing the greatest enclosed area given 200 feet of fencing.

50 feet by 50 feet. Maximize $\text{\hspace{0.17em}}f\left(x\right)=-{x}^{2}+100x.$

Find the dimensions of the rectangular corral split into 2 pens of the same size producing the greatest possible enclosed area given 300 feet of fencing.

Find the dimensions of the rectangular corral producing the greatest enclosed area split into 3 pens of the same size given 500 feet of fencing.

125 feet by 62.5 feet. Maximize $\text{\hspace{0.17em}}f\left(x\right)=-2{x}^{2}+250x.$

Among all of the pairs of numbers whose sum is 6, find the pair with the largest product. What is the product?

Among all of the pairs of numbers whose difference is 12, find the pair with the smallest product. What is the product?

$6\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}-6;\text{\hspace{0.17em}}$ product is –36; maximize $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}+12x.$

Suppose that the price per unit in dollars of a cell phone production is modeled by $\text{\hspace{0.17em}}p=45-0.0125x,\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is in thousands of phones produced, and the revenue represented by thousands of dollars is $\text{\hspace{0.17em}}R=x\cdot p.\text{\hspace{0.17em}}$ Find the production level that will maximize revenue.

A rocket is launched in the air. Its height, in meters above sea level, as a function of time, in seconds, is given by $\text{\hspace{0.17em}}h\left(t\right)=-4.9{t}^{2}+229t+234.\text{\hspace{0.17em}}$ Find the maximum height the rocket attains.

2909.56 meters

A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by $\text{\hspace{0.17em}}h\left(t\right)=-4.9{t}^{2}+24t+8.\text{\hspace{0.17em}}$ How long does it take to reach maximum height?

A soccer stadium holds 62,000 spectators. With a ticket price of $11, the average attendance has been 26,000. When the price dropped to$9, the average attendance rose to 31,000. Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue?

\$10.70

A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?

can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator