# 4.4E: Exercises - Quadratic Functions and Applications

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## Practice Makes Perfect

##### Exercises 1 - 8: Recognize the Graph of a Quadratic Function

For each of the following exercises, determine if the parabola opens up or down.

1. $$f(x)=-2 x^{2}-6 x-7$$
2. $$f(x)=6 x^{2}+2 x+3$$
3. $$f(x)=4 x^{2}+x-4$$
4. $$f(x)=-9 x^{2}-24 x-16$$
5. $$f(x)=-3 x^{2}+5 x-1$$
6. $$f(x)=2 x^{2}-4 x+5$$
7. $$f(x)=x^{2}+3 x-4$$
8. $$f(x)=-4 x^{2}-12 x-9$$

1. down

3. up

5. down

7. up

##### Exercises 9 - 12: Find the Axis of Symmetry and Vertex of a Parabola

In the following functions, find each of the following.  Round answers to the nearest tenth where necessary.

1. The vertex of its graph
2. The equation of the axis of symmetry
3. The x-intercept(s)
4. The y-intercept
5. The domain
6. The range

9. $$f(x)=x^{2}+8 x-1$$

10. $$f(x)=x^{2}+10 x+25$$

11. $$f(x)=-3x^{2}+2 x+9$$

12. $$f(x)=-2 x^{2}-12 x-3$$

9.

1. Vertex: $$(-4,-17)$$
2. Axis of symmetry: $$x=-4$$
3. x-intercepts: $$(-8.1,0), (0.1,0)$$
4. y-intercept: $$(0,-1)$$
5. Domain: $$(-\infty,\infty)$$
6. Range: $$[-17, \infty)$$

11.

1. Vertex: $$(0.3, 9.3)$$
2. Axis of symmetry: $$x=0.3$$
3. x-intercepts: $$(-1.4,0), (2.1,0)$$
4. y-intercept: $$(0,9)$$
5. Domain: $$(-\infty,\infty)$$
6. Range: $$(-\infty,9.3]$$
##### Exercises 13-18: Solve Maximum and Minimum Applications

In the following exercises, find the maximum or minimum value of each function.

13. $$f(x)=2 x^{2}+x-1$$

14. $$y=-4 x^{2}+12 x-5$$

15. $$y=x^{2}-6 x+15$$

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

17. $$y=-9 x^{2}+16$$

18. $$y=4 x^{2}-49$$

13. The minimum value is $$−\frac{9}{8}$$ when $$x=−\frac{1}{4}$$.

15. The maximum value is $$6$$ when $$x=3$$.

17. The maximum value is $$16$$ when $$x=0$$.

##### Exercises 19 - 23: Solve Maximum and Minimum Applications

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

19. A computer store owner estimates that by charging $$x$$ dollars each for a certain computer, he can sell $$40 − x$$ computers each week. The quadratic function $$R(x)=-x^{2}+40 x$$ is used to find the revenue, $$R$$, received when the selling price of a computer is $$x$$, Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.

20. A cell phone company estimates that by charging $$x$$ dollars each for a certain cell phone, they can sell $$8 − x$$ cell phones per day. Use the quadratic function $$R(x)=-x^{2}+8 x$$ to find the revenue received per day when the selling price of a cell phone is $$x$$. Find the selling price that will give them the maximum revenue per day, and then find the amount of the maximum revenue.

21. A land owner is planning to build a fenced in rectangular patio behind his garage, using his garage as one of the “walls.” He wants to maximize the area using $$80$$ feet of fencing. The quadratic function $$A(x)=x(80-2 x)$$ gives the area of the patio, where $$x$$ is the width of one side. Find the maximum area of the patio.

22. A family of three young children just moved into a house with a yard that is not fenced in. The previous owner gave them $$300$$ feet of fencing to use to enclose part of their backyard. Use the quadratic function $$A(x)=x(300-2 x)$$ to determine the maximum area of the fenced in yard.

23. A ball is thrown vertically upward from the ground with an initial velocity of $$109$$ ft/sec. Use the quadratic function $$h(t)=-16 t^{2}+109 t+0$$ to find how long it will take for the ball to reach its maximum height, and then find the maximum height.

19. $$20$$ computers will give the maximum of \$$$400$$ in receipts.
21. The maximum area of the patio is $$800$$ feet.
23. In $$3.4$$ seconds the ball will reach its maximum height of $$185.6$$ feet.