# 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? #### Questions & Answers How can you tell what type of parent function a graph is ? Mary Reply 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)= Karim Reply 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 GREAT ANSWER THOUGH!!! 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? unknown Reply 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 Ef Reply So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right? KARMEL Reply The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26 Rima Reply 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 Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that 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. Brittany Reply how do you find the period of a sine graph Imani Reply 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 Jhon Reply 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 Baptiste Reply the sum of any two linear polynomial is what Esther Reply divide simplify each answer 3/2÷5/4 Momo Reply divide simplify each answer 25/3÷5/12 Momo how can are find the domain and range of a relations austin Reply 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? Diddy Reply 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