# 10.5 Graphing quadratic equations  (Page 9/15)

 Page 9 / 15

Find two consecutive even numbers whose product is 624.

A triangular banner has an area of 351 square centimeters. The length of the base is two centimeters longer than four times the height. Find the height and length of the base.

The height of the banner is 13 cm and the length of the side is 54 cm.

Julius built a triangular display case for his coin collection. The height of the display case is six inches less than twice the width of the base. The area of the of the back of the case is 70 square inches. Find the height and width of the case.

A tile mosaic in the shape of a right triangle is used as the corner of a rectangular pathway. The hypotenuse of the mosaic is 5 feet. One side of the mosaic is twice as long as the other side. What are the lengths of the sides? Round to the nearest tenth.

The lengths of the sides of the mosaic are 2.2 and 4.4 feet.

A rectangular piece of plywood has a diagonal which measures two feet more than the width. The length of the plywood is twice the width. What is the length of the plywood’s diagonal? Round to the nearest tenth.

The front walk from the street to Pam’s house has an area of 250 square feet. Its length is two less than four times its width. Find the length and width of the sidewalk. Round to the nearest tenth.

The width of the front walk is 8.1 feet and its length is 30.8 feet.

For Sophia’s graduation party, several tables of the same width will be arranged end to end to give a serving table with a total area of 75 square feet. The total length of the tables will be two more than three times the width. Find the length and width of the serving table so Sophia can purchase the correct size tablecloth. Round answer to the nearest tenth.

A ball is thrown vertically in the air with a velocity of 160 ft/sec. Use the formula $h=-16{t}^{2}+{v}_{0}t$ to determine when the ball will be 384 feet from the ground. Round to the nearest tenth.

The ball will reach 384 feet on its way up in 4 seconds and on the way down in 6 seconds.

A bullet is fired straight up from the ground at a velocity of 320 ft/sec. Use the formula $h=-16{t}^{2}+{v}_{0}t$ to determine when the bullet will reach 800 feet. Round to the nearest tenth.

## 10.5 Graphing Quadratic Equations in Two Variables

In the following exercises, graph by plotting point.

Graph $y={x}^{2}-2$

Graph $y=\text{−}{x}^{2}+3$

In the following exercises, determine if the following parabolas open up or down.

$y=-3{x}^{2}+3x-1$

down

$y=5{x}^{2}+6x+3$

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

up

$y=-4{x}^{2}-7x+1$

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

$y=\text{−}{x}^{2}+6x+8$

$x=3$ $\left(3,17\right)$

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

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

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

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

$y={x}^{2}-8x+15$

$y={x}^{2}-4x+10$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,10\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\text{none}$

$y=-5{x}^{2}-30x-46$

$y=16{x}^{2}-8x+1$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,1\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\left(\frac{1}{4},0\right)$

$y={x}^{2}+16x+64$

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

$y={x}^{2}+8x+15$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,15\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-3,0\right),\left(-5,0\right);$
axis: $x=-4;\text{vertex}\text{:}\phantom{\rule{0.2em}{0ex}}\left(-4,-1\right)$

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

$y=\text{−}{x}^{2}+8x-16$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,-16\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\left(4,0\right);$
axis: $x=4;\text{vertex}\text{:}\phantom{\rule{0.2em}{0ex}}\left(4,0\right)$

$y=4{x}^{2}-4x+1$

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

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,13\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\text{none};$
axis: $x=-3;\text{vertex}\text{:}\phantom{\rule{0.2em}{0ex}}\left(-3,4\right)$

$y=-2{x}^{2}-8x-12$

$y=-4{x}^{2}+16x-11$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,-11\right)\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\left(3.1,0\right),\left(0.9,0\right);$
axis: $x=2;\phantom{\rule{0.2em}{0ex}}\text{vertex:}\phantom{\rule{0.2em}{0ex}}\left(2,5\right)$

$y={x}^{2}+8x+10$

In the following exercises, find the minimum or maximum value.

$y=7{x}^{2}+14x+6$

The minimum value is $-1$ when $x=-1$ .

$y=-3{x}^{2}+12x-10$

In the following exercises, solve. Rounding answers to the nearest tenth.

A ball is thrown upward from the ground with an initial velocity of 112 ft/sec. Use the quadratic equation $h=-16{t}^{2}+112t$ to find how long it will take the ball to reach maximum height, and then find the maximum height.

In 3.5 seconds the ball is at its maximum height of 196 feet.

A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using 180 feet of fencing on three sides of the yard. The quadratic equation $A=-2{x}^{2}+180x$ gives the area, $A$ , of the yard for the length, $x$ , of the building that will border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximum area.

## Practice test

Use the Square Root Property to solve the quadratic equation: $3{\left(w+5\right)}^{2}=27$ .

$w=-2,-8$

Use Completing the Square to solve the quadratic equation: ${a}^{2}-8a+7=23$ .

Use the Quadratic Formula to solve the quadratic equation: $2{m}^{2}-5m+3=0$ .

$m=1,\frac{3}{2}$

Solve the following quadratic equations. Use any method.

$8{v}^{2}+3=35$

$3{n}^{2}+8n+3=0$

$n=\frac{-4±\sqrt{7}}{3}$

$2{b}^{2}+6b-8=0$

$x\left(x+3\right)+12=0$

no real solution

$\frac{4}{3}{y}^{2}-4y+3=0$

Use the discriminant to determine the number of solutions of each quadratic equation.

$6{p}^{2}-13p+7=0$

2

$3{q}^{2}-10q+12=0$

Solve by factoring, the Square Root Property, or the Quadratic Formula.

Find two consecutive even numbers whose product is 360.

Two consecutive even number are $-20$ and $-18$ and 18 and 20.

The length of a diagonal of a rectangle is three more than the width. The length of the rectangle is three times the width. Find the length of the diagonal. (Round to the nearest tenth.)

For each parabola, find which ways it opens, the axis of symmetry, the vertex, the x - and y -intercepts, and the maximum or minimum value.

$y=3{x}^{2}+6x+8$

up $x=-1$ $\left(-1,5\right)$ $y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,8\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\text{none}$ minimum value of 5 when $x=-1$

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

$y={x}^{2}+10x+24$

up $x=-5$ $\left(-5,-1\right)$ $y;\left(0,24\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-6,0\right),\left(-4,0\right)$ minimum value of $-5$ when $x=-1$

$y=-3{x}^{2}+12x-8$

$y=\text{−}{x}^{2}-8x+16$

down $x=-4$
$\left(-4,32\right)$ $y;\left(0,16\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-9.7,0\right),\left(1.7,0\right)$
maximum value of $32$ when $x=-4$

Graph the following parabolas by using intercepts, the vertex, and the axis of symmetry.

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

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

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,9\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-\frac{3}{4},0\right);$
axis: $x=-\frac{3}{4};\text{vertex}\text{:}\phantom{\rule{0.2em}{0ex}}\left(-\frac{3}{4},0\right)$

Solve.

A water balloon is launched upward at the rate of 86 ft/sec. Using the formula $h=-16{t}^{2}+86t$ , find how long it will take the balloon to reach the maximum height and then find the maximum height. Round to the nearest tenth.

what is nanomaterials​ and their applications of sensors.
what is nano technology
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
app is wrong how can 350 be divisible by 3.
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
I'm getting "math processing error" on math problems. Anyone know why?
Can you all help me I don't get any of this
4^×=9
Did anyone else have trouble getting in quiz link for linear inequalities?
operation of trinomial
y=2×+9
Keshad gets paid $2,400 per month plus 6% of his sales. His brother earns$3,300 per month. For what amount of total sales will Keshad’s monthly pay be higher than his brother’s monthly pay?
Mayra has $124 in her checking account. She writes a check for$152. What is the New Balance in her checking account?