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College Algebra 12th

R. David Gustafson, Jeff Hughes

Chapter 4

Polynomial and Rational Functions

Educators


Problem 1

Fill in the blanks.
A quadratic function is defined by the equation _______ $(a \neq 0)$.

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Problem 2

Fill in the blanks.
The standard form for the equation of a parabola is ______ $(a \neq 0)$

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Problem 3

Fill in the blanks.
The vertex of the parabolic graph of the equation $y=2(x-3)^{2}+5$ will be at _________ .

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Problem 4

Fill in the blanks.
The vertical line that intersects the parabola at its vertex is the _________ .

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Problem 5

Fill in the blanks.
If the parabola opens _______ the vertex will be a minimum point.

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Problem 6

Fill in the blanks.
If the parabola opens ______ the vertex will be a maximum point.

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Problem 7

Fill in the blanks.
The $x$ -coordinate of the vertex of the parabolic graph of $f(x)=a x^{2}+b x+c$ is ______.

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Problem 8

Fill in the blanks.
The $y$ -coordinate of the vertex of the parabolic graph of $f(x)=a x^{2}+b x+c$ is __________.

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Problem 9

Determine whether the graph of each quadratic function opens upward or downward. State whether a
maximum or minimum point occurs at the vertex of the parabola.
$$f(x)=\frac{1}{2} x^{2}+3$$

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Problem 10

Determine whether the graph of each quadratic function opens upward or downward. State whether a
maximum or minimum point occurs at the vertex of the parabola.
$$f(x)=2 x^{2}-3 x$$

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Problem 11

Determine whether the graph of each quadratic function opens upward or downward. State whether a
maximum or minimum point occurs at the vertex of the parabola.
$$f(x)=-3(x+1)^{2}+2$$

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Problem 12

Determine whether the graph of each quadratic function opens upward or downward. State whether a
maximum or minimum point occurs at the vertex of the parabola.
$$f(x)=-5(x-1)^{2}-1$$

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Problem 13

Determine whether the graph of each quadratic function opens upward or downward. State whether a
maximum or minimum point occurs at the vertex of the parabola.
$$f(x)=-2 x^{2}+5 x-1$$

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Problem 14

Determine whether the graph of each quadratic function opens upward or downward. State whether a
maximum or minimum point occurs at the vertex of the parabola.
$$f(x)=2 x^{2}-3 x+1$$

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Problem 15

Find the vertex of each parabola.
$$f(x)=x^{2}-1$$

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Problem 16

Find the vertex of each parabola.
$$f(x)=-x^{2}+2$$

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Problem 17

Find the vertex of each parabola.
$$f(x)=(x-3)^{2}+5$$

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Problem 18

Find the vertex of each parabola.
$$f(x)=-2(x-3)^{2}+4$$

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Problem 19

Find the vertex of each parabola.
$$f(x)=-2(x+6)^{2}-4$$

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Problem 20

Find the vertex of each parabola.
$$f(x)=\frac{1}{3}(x+1)^{2}-5$$

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Problem 21

Find the vertex of each parabola.
$$f(x)=\frac{2}{3}(x-3)^{2}$$

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Problem 22

Find the vertex of each parabola.
$$f(x)=7(x+2)^{2}+8$$

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Problem 23

Find the vertex of each parabola.
$$f(x)=x^{2}-4 x+4$$

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Problem 24

Find the vertex of each parabola.
$$f(x)=x^{2}-10 x+25$$

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Problem 25

Find the vertex of each parabola.
$$f(x)=x^{2}+6 x-3$$

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Problem 26

Find the vertex of each parabola.
$$f(x)=-x^{2}+9 x-2$$

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Problem 27

Find the vertex of each parabola.
$$f(x)=-2 x^{2}+12 x-17$$

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Problem 28

Find the vertex of each parabola.
$$f(x)=2 x^{2}+16 x+33$$

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Problem 29

Find the vertex of each parabola.
$$f(x)=3 x^{2}-4 x+5$$

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Problem 30

Find the vertex of each parabola.
$$f(x)=-4 x^{2}+3 x+4$$

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Problem 31

Find the vertex of each parabola.
$$f(x)=\frac{1}{2} x^{2}+4 x-3$$

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Problem 32

Find the vertex of each parabola.
$$f(x)=-\frac{2}{3} x^{2}+3 x-5$$

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Problem 33

Graph each quadratic function given in standard form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=x^{2}-4$$

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Problem 34

Graph each quadratic function given in standard form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=x^{2}+1$$

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Problem 35

Graph each quadratic function given in standard form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=-3 x^{2}+6$$

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Problem 36

Graph each quadratic function given in standard form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=-4 x^{2}+4$$

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Problem 37

Graph each quadratic function given in standard form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=-\frac{1}{2} x^{2}+8$$

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Problem 38

Graph each quadratic function given in standard form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=\frac{1}{2} x^{2}-2$$

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Problem 39

Graph each quadratic function given in standard form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=(x-3)^{2}-1$$

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Problem 40

Graph each quadratic function given in standard form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=(x+3)^{2}-1$$

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Problem 41

Graph each quadratic function given in standard form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=2(x+1)^{2}-2$$

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Problem 42

Graph each quadratic function given in standard form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=-\frac{3}{4}(x-2)^{2}$$

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Problem 43

Graph each quadratic function given in standard form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=-(x+4)^{2}+1$$

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Problem 44

Graph each quadratic function given in standard form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=-3(x-4)^{2}+3$$

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Problem 45

Graph each quadratic function given in standard form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=-3(x-2)^{2}+6$$

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Problem 46

Graph each quadratic function given in standard form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=2(x-3)^{2}-4$$

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Problem 47

Graph each quadratic function given in standard form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=\frac{1}{3}(x-1)^{2}-3$$

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Problem 48

Graph each quadratic function given in standard form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=-\frac{1}{2}(x+1)^{2}+8$$

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Problem 49

Graph each quadratic function given in general form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=x^{2}+2 x$$

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Problem 50

Graph each quadratic function given in general form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=x^{2}-6 x$$

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Problem 51

Graph each quadratic function given in general form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=x^{2}-6 x-7$$

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Problem 52

Graph each quadratic function given in general form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=x^{2}-4 x+1$$

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Problem 53

Graph each quadratic function given in general form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=-x^{2}-4 x+1$$

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Problem 54

Graph each quadratic function given in general form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=-x^{2}-x+6$$

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Problem 55

Graph each quadratic function given in general form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=2 x^{2}-12 x+10$$

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Problem 56

Graph each quadratic function given in general form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=-3 x^{2}-3 x+18$$

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Problem 57

Graph each quadratic function given in general form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=-3 x^{2}-6 x-9$$

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Problem 58

Graph each quadratic function given in general form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=-4 x^{2}-4 x+3$$

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Problem 59

Graph each quadratic function given in general form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=\frac{1}{2} x^{2}-2 x-\frac{5}{2}$$

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Problem 60

Graph each quadratic function given in general form. Identify the vertex, intercepts, and axis of
symmetry.
$$f(x)=-\frac{1}{2} x^{2}-x+4$$

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Problem 61

A police officer seals off the scene of an accident using a roll of yellow tape that is 300 feet long. What dimensions should be used to seal off the maximum rectangular area around the collision? Find the maximum area.

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Problem 62

A rectangular flower bed has a width of $x$ feet and a perimeter of 100 feet. Find $x$ such that the area of the rectangle is maximized.

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Problem 63

Jake has 800 feet of fencing to enclose a rectangular plot of land that borders a river. If Jake doesn't need a fence along the side of the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

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Problem 64

A rectangular parking lot is being constructed for your college football stadium. If the parking lot is bordered on one side by a street and there are 750 yards of fencing available for the other three sides, find the length and width of the lot that will maximize the area. What is the largest area that can be enclosed?

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Problem 65

A farmer wants to partition a rectangular feed storage area in a corner of his barn, as shown in the illustration. The barn walls form two sides of the stall, and the farmer has 50 feet of partition for the remaining two sides. What dimensions will maximize the area?

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Problem 66

A rancher wishes to enclose a rectangular partitioned corral with 1800 feet of fencing. (See the illustration.) What dimensions of the corral would enclose the largest possible area? Find the maximum area.

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Problem 67

$A 24$ -inch-wide sheet of metal is to be bent into a rectangular trough with the cross section shown in the illustration. Find the dimensions that will maximize the amount of water the trough can hold. That is, find the dimensions that will maximize the cross-sectional area.

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Problem 68

A 90 -foot-wide sheet of metal is to be bent to form a rectangular trough from which your animals will drink water. Find the dimensions that will maximize the amount of water the trough can hold. That is, find the cross-sectional area of the trough.

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Problem 69

A parabolic arch has an equation of $x^{2}+20 y-400=0,$ where $x$ is measured in feet. Find the maximum height of the arch.

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Problem 70

A guided missile is propelled from the origin of a coordinate system with the $x$ -axis along the ground and the $y$ -axis vertical. Its path, or trajectory, is given by the equation $y=400 x-16 x^{2} .$ Find the object's maximum height.

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Problem 71

The path of a basketball thrown from the free throw line can be modeled by the quadratic function $f(x)=-0.06 x^{2}+1.5 x+6,$ where $x$ is the horizontal distance (in feet) from the free throw line and $f(x)$ is the height (in feet) of the ball. Find the maximum height of the basketball.

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Problem 72

Projectile motion Devin throws a ball up a hill that makes an angle of $45^{\circ}$ with the horizontal. The ball lands 100 feet up the hill. Its trajectory is a parabola with equation $y=-x^{2}+a x$ for some number a. Find $a$.

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Problem 73

A football is thrown by a quarterback from the 10 -yard line and caught by the wide receiver on the 50 -yard line. The football's path on this interval can be modeled by the quadratic function $f(x)=-\frac{1}{20} x^{2}+3 x-19,$ where $x$ is the horizontal distance in yards from the goal line and $f(x)$ is the height of the football in feet. Find the maximum height reached by the football.

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Problem 74

A ball is thrown straight up from the top of a building 144 ft. tall with an initial velocity of $64 \mathrm{ft}$ per second. The height $s(t)$ (in feet) of the ball from the ground, at time $t$ (in seconds), is given by $s(t)=144+64 t-16 t^{2} .$ Find the maximum height attained by the ball.

Sheryl E.
Numerade Educator

Problem 75

A wholesaler of appliances finds that she can sell (1200 - $x$ ) flat-screen television sets each week when the price is $x$ dollars. What price will maximize revenue?

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Problem 76

A seller of contemporary desks finds that he can sell $(820-x)$ desks each month when the price is $x$ dollars. What price will maximize revenue?

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Problem 77

A company that produces and sells digital cameras has determined that the total weekly cost $C(x),$ in dollars, of producing $x$ digital cameras is given by the function $C(x)=1.5 x^{2}-144 x+5856 .$ Determine the production level that minimizes the weekly cost for producing the digital cameras and find that weekly minimum cost.

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Problem 78

A company that produces and sells chandeliers has determined that the total monthly profit $P(x)$ in dollars of producing and selling $x$ chandeliers is given by the function $P(x)=-1.5 x^{2}+153 x+7215 .$ Determine the production level that maximizes the monthly profit, and find that maximum profit.

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Problem 79

The Municipal Transit Authority serves 150,000 commuters daily when the fare is $\$ 1.80 .$ Market research has determined that every penny decrease in the fare will result in 1000 new riders. What fare will maximize revenue?

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Problem 80

Tickets for a concert are cheaper when purchased in quantity. The first 100 tickets are priced at $\$ 10$ each, but each additional block of 100 tickets purchased decreases the cost of each ticket by 50c. How many blocks of tickets should be sold to maximize the revenue?

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Problem 81

$A 300$ -room hotel is two-thirds filled when the nightly room rate is $\$ 90 .$ Experience has shown that each $\$ 5$ increase in cost results in 10 fewer occupied rooms. Find the nightly rate that will maximize income.

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Problem 82

A 500 -room Hilton hotel is $80 \%$ filled when the nightly room rate is $\$ 160$. Experience has shown that each $\$ 5$ increase in the rate results in 10 fewer occupied rooms. Find the nightly rate that will maximize the nightly revenue.

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Problem 83

An object is tossed vertically upward from ground level. Its height $s(t),$ in feet, at time $t$ seconds is given by the position function $s(t)=-16 t^{2}+80 t$.
In how many seconds does the object reach its maximum height?

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Problem 84

An object is tossed vertically upward from ground level. Its height $s(t),$ in feet, at time $t$ seconds is given by the position function $s(t)=-16 t^{2}+80 t$.
In how many seconds does the object return to the point from which it was thrown?

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Problem 85

An object is tossed vertically upward from ground level. Its height $s(t),$ in feet, at time $t$ seconds is given by the position function $s(t)=-16 t^{2}+80 t$.
What is the maximum height reached by the object?

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Problem 86

An object is tossed vertically upward from ground level. Its height $s(t),$ in feet, at time $t$ seconds is given by the position function $s(t)=-16 t^{2}+80 t$.
Show that it takes the same amount of time for the object to reach its maximum height as it does to return from that height to the point from which it was thrown.

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Problem 87

Use a graphing calculator to determine the coordinates of the vertex of each parabola. You will have to select appropriate viewing windows.
$$y=2 x^{2}+9 x-56$$

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Problem 88

Use a graphing calculator to determine the coordinates of the vertex of each parabola. You will have to select appropriate viewing windows.
$$y=14 x-\frac{x^{2}}{5}$$

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Problem 89

Use a graphing calculator to determine the coordinates of the vertex of each parabola. You will have to select appropriate viewing windows.
$$y=(x-7)(5 x+2)$$

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Problem 90

Use a graphing calculator to determine the coordinates of the vertex of each parabola. You will have to select appropriate viewing windows.
$$y=-x(0.2+0.1 x)$$

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Problem 91

Use a graphing calculator and quadratic regression to find the quadratic function that best fits the given set of data.
$\{(-1,6),(0,-1),(1,-3),(2,-1.5),(3,5),(4,10)\}$ Round to three decimal places.

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Problem 92

Use a graphing calculator and quadratic regression to find the quadratic function that best fits the given set of data.
$\{(-3,4),(-1,5),(0,7),(2,9),(5,7),(6,5)\}$ Round to three decimal places.

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Problem 93

The length (in inches) and weight (in pounds) of 25 alligators is shown in the table. Find "me quadratic function that best fits the data. Round $a, b,$ and $c$ to six decimal places. Use the regression function to estimate the weight of an alligator that is 130 inches long. Round the weight "to the nearest pound.

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Problem 94

Refer to Exercise $93 .$ If an alligator weighs 125 pounds, what is its approximate length? Round to the nearest inch.

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Problem 95

What is a quadratic function?

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Problem 96

Describe two ways of finding the vertex of a parabola given in general form.

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Problem 97

What is an axis of symmetry of a parabola?

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Problem 98

Share the strategy you would use to solve a maximum or minimum application problem.

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Problem 99

Find the dimensions of the largest rectangle that can be inscribed in the right triangle $A B C$ shown in the illustration.

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Problem 100

Point $P$ lies in the first quadrant and on the line $x+y=1$ in such a position that the area of triangle OPA is maximum. Find the coordinates of $P$. (See the illustration.)

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Problem 101

The sum of two numbers is $6,$ and the sum of the squares of those two numbers is as small as
possible. What are the numbers?

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Problem 102

What number most exceeds its square?

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Problem 103

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
If the sum of a number and its square is a minimum, then the two numbers are $-\frac{1}{2}$ and $\frac{1}{4}$.

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Problem 104

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
The graphs of some quadratic functions have no $x$ -intercepts.

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Problem 105

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
The graphs of some quadratic functions have no $x$ -intercepts.

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Problem 106

The graph of a quadratic function is never constant.

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Problem 107

If $f(x)=a(x-h)^{2}+k,$ and $a>0,$ then the range is $(-\infty, k]$

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Problem 108

If $f(x)=a(x-h)^{2}+k,$ and $a>0,$ then the range is $(-\infty, k]$.

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Problem 109

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
If $g(x)=a x^{2}+b x+c,$ and $a>0,$ then the graph of the function is increasing on $\left[-\frac{b}{2 a}, \infty\right)$

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Problem 110

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
The axis of symmetry of the parabola $f(x)=444 x^{2}-888 x+222$ is $x=-1$

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