# 10.5 Graphing quadratic equations  (Page 6/15)

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The quadratic equation $h=-16{t}^{2}+128t+32$ is used to find the height of a stone thrown upward from a height of 32 feet at a rate of 128 ft/sec. How long will it take for the stone to reach its maximum height? What is the maximum height? Round answers to the nearest tenth.

It will take 4 seconds to reach the maximum height of 288 feet.

A toy rocket shot upward from the ground at a rate of 208 ft/sec has the quadratic equation of $h=-16{t}^{2}+208t$ . When will the rocket reach its maximum height? What will be the maximum height? Round answers to the nearest tenth.

It will take 6.5 seconds to reach the maximum height of 676 feet.

Access these online resources for additional instruction and practice graphing quadratic equations:

## Key concepts

• The graph of every quadratic equation is a parabola.
• Parabola Orientation For the quadratic equation $y=a{x}^{2}+bx+c$ , if
• $a>0$ , the parabola opens upward.
• $a<0$ , the parabola opens downward.
• Axis of Symmetry and Vertex of a Parabola For a parabola with equation $y=a{x}^{2}+bx+c$ :
• The axis of symmetry of a parabola is the line $x=-\frac{b}{2a}$ .
• The vertex is on the axis of symmetry, so its x -coordinate is $-\frac{b}{2a}$ .
• To find the y -coordinate of the vertex we substitute $x=-\frac{b}{2a}$ into the quadratic equation.
• Find the Intercepts of a Parabola To find the intercepts of a parabola with equation $y=a{x}^{2}+bx+c$ :
$\begin{array}{cccc}\hfill {\text{y}}\mathbf{\text{-intercept}}& & & \hfill {\text{x}}\mathbf{\text{-intercepts}}\\ \hfill \text{Let}\phantom{\rule{0.2em}{0ex}}x=0\phantom{\rule{0.2em}{0ex}}\text{and solve for}\phantom{\rule{0.2em}{0ex}}y.& & & \hfill \text{Let}\phantom{\rule{0.2em}{0ex}}y=0\phantom{\rule{0.2em}{0ex}}\text{and solve for}\phantom{\rule{0.2em}{0ex}}x.\end{array}$
• To Graph a Quadratic Equation in Two Variables
1. Write the quadratic equation with $y$ on one side.
2. Determine whether the parabola opens upward or downward.
3. Find the axis of symmetry.
4. Find the vertex.
5. Find the y -intercept. Find the point symmetric to the y -intercept across the axis of symmetry.
6. Find the x -intercepts.
7. Graph the parabola.
• Minimum or Maximum Values of a Quadratic Equation
• The y - coordinate of the vertex of the graph of a quadratic equation is the
• minimum value of the quadratic equation if the parabola opens upward.
• maximum value of the quadratic equation if the parabola opens downward.

## Practice makes perfect

Recognize the Graph of a Quadratic Equation in Two Variables

In the following exercises, graph:

$y={x}^{2}+3$

$y=\text{−}{x}^{2}+1$

In the following exercises, determine if the parabola opens up or down.

$y=-2{x}^{2}-6x-7$

down

$y=6{x}^{2}+2x+3$

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

up

$y=-9{x}^{2}-24x-16$

Find the Axis of Symmetry and Vertex of a Parabola

In the following exercises, find the axis of symmetry and the vertex.

$y={x}^{2}+8x-1$

$x=-4$ $\left(-4,-17\right)$

$y={x}^{2}+10x+25$

$y=\text{−}{x}^{2}+2x+5$

$x=1$ $\left(1,6\right)$

$y=-2{x}^{2}-8x-3$

Find the Intercepts of a Parabola

In the following exercises, find the x - and y -intercepts.

$y={x}^{2}+7x+6$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,6\right);\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-1,0\right),\left(-6,0\right)$

$y={x}^{2}+10x-11$

$y=\text{−}{x}^{2}+8x-19$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,19\right);\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\text{none}$

$y={x}^{2}+6x+13$

$y=4{x}^{2}-20x+25$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,25\right);\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\left(\frac{5}{2},0\right)$

$y=\text{−}{x}^{2}-14x-49$

Graph Quadratic Equations in Two Variables

In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry.

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

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,5\right);\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-1,0\right),\left(-5,0\right);$
axis: $x=-3;\phantom{\rule{0.2em}{0ex}}\text{vertex}\text{:}\phantom{\rule{0.2em}{0ex}}\left(-3,-4\right)$

$y={x}^{2}+4x-12$

$y={x}^{2}+4x+3$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,3\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-1,0\right),\left(-3,0\right);$
axis: $x=-2;\phantom{\rule{0.2em}{0ex}}\text{vertex:}\phantom{\rule{0.2em}{0ex}}\left(-2,-1\right)$

$y={x}^{2}-6x+8$

$y=9{x}^{2}+12x+4$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,4\right)\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-\frac{2}{3},0\right);$
axis: $x=-\frac{2}{3};\phantom{\rule{0.2em}{0ex}}\text{vertex:}\phantom{\rule{0.2em}{0ex}}\left(-\frac{2}{3},0\right)$

$y=\text{−}{x}^{2}+8x-16$

$y=\text{−}{x}^{2}+2x-7$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,-7\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\text{none};$
axis: $x=1;\text{vertex}\text{:}\phantom{\rule{0.2em}{0ex}}\left(1,-6\right)$

integer greater than 2 and less than 12
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
He charges $125 per job. His monthly expenses are$1,600. How many jobs must he work in order to make a profit of at least $2,400? Alicia Reply at least 20 Ayla what are the steps? Alicia 6.4 jobs Grahame 32 Grahame 1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32 Orlando He must work 32 jobs to make a profit POP what is algebra Azhar Reply repeated addition and subtraction of the order of operations. i love algebra I'm obsessed. Shemiah hi Krekar One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag? Leanna Reply they are 92 candies in the bag POP rectangular field solutions Navin Reply What is this? Donna t muqtaar the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is anas Reply ? Choli a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190 Usman Stella bought a dinette set on sale for$725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount? Manhwa Reply 44.19% Scott 40.22% Terence 44.2% Orlando I don't know Donna if you want the discounted price subtract$725 from $1299. then divide the answer by$1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando
p mulripied-5 and add 30 to it
Tausif
Tausif
how
muqtaar
Can you explain further
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob
How do you find divisible numbers without a calculator?
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was$1,585. What is the amount of each loan?
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
No; 65m/hr
albert
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
what do you need help in?
Felix
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found$7(+7). Totally you lost just $12 (-12) Annushka I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-\$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
hi
albert
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce