Q: The average cost of tuition and fees at public colleges, $$T$$, for the school years ending $$x$$ years after 2000 is modeled by $$T = -x^2$ + 325x + 3101. $$According to this model, in which year will the average cost of tuition and fees at public colleges reach a maximum?

$$x = 162.5$$; \\text{Year: } 2000 + 162.5 = 2162.5$

Question 1

The average cost of tuition and fees at public colleges, $$T$$, for the school years ending $$x$$ years after 2000 is modeled by $$T = -x^2$ + 325x + 3101. $$According to this model, in which year will the average cost of tuition and fees at public colleges reach a maximum?

Accepted Answer

$$x = 162.5$$; \text{Year: } 2000 + 162.5 = 2162.5$

Question 2

Solve the radical equation: $$\sqrt{2x + 5} + 5 = x. $$Which of the following is a valid solution?

Accepted Answer

$$x = 6$$

Question 3

Solve the quadratic equation: $$2x^2 + 7x = 1$. What are the solutions?

Accepted Answer

$$x = \frac{-7 \pm \sqrt{57}}{4}$$

Question 4

Solve the quadratic equation: $$3x^2 + 3x - 3 = 0$. Which of the following is correct?

Accepted Answer

$$x = \frac{-3 \pm \sqrt{3^2$ - 4 \cdot 3 \cdot (-3)}}{2 \cdot 3}$$

Question 5

Solve the quadratic equation: $$12x^2 + 24x = 12$. What are the solutions?

Accepted Answer

$$x = -1 \pm \frac{\sqrt{2}}{2}$$

Question 6

Solve the quadratic equation: $$x^2 = 2x + 3$. What are the solutions?

Accepted Answer

Question 7

Compute the discriminant, then solve the quadratic equation: $$2x^2 + 7x - 14 = 0$. What is the discriminant?

Accepted Answer

$$D = 7^2$ - 4 \cdot 2 \cdot (-14) = 137$$

Question 8

Solve the quadratic equation by factoring or completing the square: $$x^2 = 5 - 4x$. What are the solutions?

Accepted Answer

$$x = -2 \pm 3$$

Question 9

Given $$f(x) = 3x^2$ - 4x - 8$$, what is the vertex of the parabola?

Accepted Answer

$$(\frac{2}{3}, -\frac{40}{3})$$

Question 10

Given $$f(x) = 3x^2$ - 4x - 8$$, what is the equation for the axis of symmetry?

Accepted Answer

$$x = \frac{2}{3}$$

Question 11

Given $$f(x) = 3x^2$ - 4x - 8$$, what is the $$y-$$intercept?

Accepted Answer

$$(0, -8)$$

Question 12

Given $$f(x) = 3x^2$ - 4x - 8$$, what are the $$x-$$intercepts?

Accepted Answer

$$(\frac{4 \pm \sqrt{16 + 96}}{6}, 0)$$

Question 13

Given $$f(x) = 3x^2$ - 4x - 8$$, what is the domain?

Accepted Answer

$$(-\infty, \infty)$$

Question 14

Given $$f(x) = 3x^2$ - 4x - 8$$, what is the range?

Accepted Answer

$$[-\frac{40}{3}, \infty)$$

Question 15

A ball is thrown upward and outward from a height of 7 feet. The height of the ball, $$f(x) in $$feet, is modeled by $$f(x) = -0.1x^2$ + 0.8x + 7$$, where $$x is $$the ball’s horizontal distance in feet. What is the maximum height of the ball and how far from where it was thrown does this occur?

Accepted Answer

Maximum height is $$8.6$$ feet at $$x = 4$$ feet

Question 16

A ball is thrown upward and outward from a height of 7 feet. The height of the ball, $$f(x) in $$feet, is modeled by $$f(x) = -0.1x^2$ + 0.8x + 7. $$How far does the ball travel horizontally before hitting the ground?

Accepted Answer

$$x = 10.7$$ feet (rounded to the nearest tenth)

Question 17

Consider the function $$h(x) = |x - 5| - 8. $$What is the $$y-$$intercept?

Accepted Answer

$$(0, -3)$$

Question 18

Consider the function $$h(x) = |x - 5| - 8. $$What are the $$x-$$intercepts?

Accepted Answer

$$(13, 0)$$ and $$(-3, 0)$$

Question 19

Consider the function $$h(x) = |x - 5| - 8. $$What is $$h(2)$$?

Accepted Answer

$$-11$$

Question 20

Consider the function $$h(x) = |x - 5| - 8. $$What is the range?

Accepted Answer

$$[-8, \infty)$$

Precalculus Study Guide: Quadratic, Radical, and Absolute Value Equations

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