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

No. 3^32 -1 has exactly two divisors greater than 75 and less than 85 what is their product?
x^2+7x-19=0 has Two solutions A and B give your answer to 3 decimal places
3. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes onthe elliptical trainer and 30 minutes circuit training she burned 473 calories. How manycalories does she burn for each minute on the elliptical trainer? How many calories doesshe burn for each minute of circuit training?
.473
Angelita
?
Angelita
John left his house in Irvine at 8:35 am to drive to a meeting in Los Angeles, 45 miles away. He arrived at the meeting at 9:50. At 3:30 pm, he left the meeting and drove home. He arrived home at 5:18.
p-2/3=5/6 how do I solve it with explanation pls
P=3/2
Vanarith
1/2p2-2/3p=5p/6
James
Cindy
4.5
Ruth
is y=7/5 a solution of 5y+3=10y-4
yes
James
Cindy
Lucinda has a pocketful of dimes and quarters with a value of $6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have? Rhonda Reply Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 . Gabriel Reply is that a negative 4 or positive 4? Felix y = mx + b Felix if negative -4, then -4=8/9(0) + b Felix -4=b Felix if positive 4, then 4=b Felix then plug in y=8/9x - 4 or y=8/9x+4 Felix Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost$9 per pound and almonds cost $5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost$6.50 per pound to make?
Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350 % mark - up Makaila Reply$1750
Cindy
the sum of two Numbers is 19 and their difference is 15
2, 17
Jose
interesting
saw
4,2
Cindy
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
hola saben como aser un valor de la expresión
NAILEA
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
hi
Darianna
hello
Mister
Eric here
Eric
6
Cindy
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?
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
32
Cindy