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

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

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

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

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

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

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

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

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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$A 24$ -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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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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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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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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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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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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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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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

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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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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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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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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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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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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$A 300$ -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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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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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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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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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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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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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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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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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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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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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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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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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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Refer to Exercise $93 .$ If an alligator weighs 125 pounds, what is its approximate length? Round to the nearest inch.

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

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

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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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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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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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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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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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The graphs of some quadratic functions have no $x$ -intercepts.

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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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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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