Q: A ball is projected upward from a height of 100 ft with an initial velocity of 80 ft/s. The height after $$t$$ seconds is given by $$h(t) = -16t^2$ + 80t + 100. $$After how many seconds does the ball reach its maximum height?

$$t = -\frac{b}{2a} = -\frac{80}{2 \times (-16)} = 2.5$$

Q: If $$f(x) = x^2$ - 4x + 7$$, what is the vertex of the parabola?

$$\left(2, 2^2$ - 4 \times 2 + 7\right) = (2, 3)$$

Question 1

Which of the following ordered pairs correctly represents the relationship between the category 'housing' and the amount spent, based on the table provided?

Accepted Answer

$$(	ext{housing},\ 21374)$$

Question 2

Given the points $$A(8, 4)$$ and $$B(3, -2)$$, what is the distance between them using the distance formula?

Accepted Answer

$$\sqrt{(8-3)^2$ + (4-(-2))^2$} = \sqrt{25 + 36} = \sqrt{61}$$

Question 3

If the midpoint of a segment with endpoints $$(8, -4)$$ and $$(-6, 1) is$$ $$M$$, what are the coordinates of $$M$$?

Accepted Answer

$$\left(\frac{8 + (-6)}{2}, \frac{-4 + 1}{2}\right) = (1, -1.5)$$

Question 4

Which of the following best describes the domain and range of the function $$f(x) = 2x + 6$$?

Accepted Answer

Domain: $$(-\infty, \infty)$$, Range: $$(-\infty, \infty)$$

Question 5

Which of the following relations is a function?

Accepted Answer

$$\{(1,2), (2,4), (3,4)\}$$

Question 6

Given the function $$f(x) = x^2$, which of the following statements is true?

Accepted Answer

$$f(x) is $$even and its graph is symmetric with respect to the y-axis.

Question 7

A ball is projected upward from a height of 100 ft with an initial velocity of 80 ft/s. The height after $$t$$ seconds is given by $$h(t) = -16t^2$ + 80t + 100. $$After how many seconds does the ball reach its maximum height?

Accepted Answer

$$t = -\frac{b}{2a} = -\frac{80}{2 	imes (-16)} = 2.5$$

Question 8

Which equation represents a vertical line passing through the point $$(3, -2)$$?

Accepted Answer

$$x = 3$$

Question 9

If $$f(x) = x^2$ - 4x + 7$$, what is the vertex of the parabola?

Accepted Answer

$$\left(2, 2^2$ - 4 	imes 2 + 7ight) = (2, 3)$$

Question 10

Given $$f(x) = x + 2$$ and $$g(x) = x^2$, what is (f$$ \circ g)(1)$$?

Accepted Answer

$$f(g(1)) = f(1^2$) = f(1) = 1 + 2 = 3$$

Question 11

Which of the following best describes the effect of the transformation $$y = f(x) + 3 on $$the graph of $$y = f(x)$$?

Accepted Answer

It shifts the graph 3 units upward.

Question 12

Which of the following is the correct formula for the slope $$m of a $$line passing through points $$(x_1$, y_1$)$$ and $$(x_2$, y_2$)$$?

Accepted Answer

$$m = \frac{y_2$ - y_1$}{x_2$ - x_1$}$$

Question 13

Given the function $$f(x) = |x|$$, which of the following statements is true?

Accepted Answer

$$f(x) is $$even and its graph is symmetric with respect to the y-axis.

Question 14

If $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$, what is the domain of (f$$ \circ g)(x)$$?

Accepted Answer

$$[0, \infty)$$

Question 15

A company’s cost function is $$C(x) = 100x + 1500$$ and the revenue function is $$R(x) = 125x. $$What is the profit function $$P(x)$$?

Accepted Answer

$$P(x) = R(x) - C(x) = 125x - (100x + 1500) = 25x - 1500$$

Question 16

Which of the following best describes the graph of $$y = x^2$?

Accepted Answer

A parabola opening upward with vertex at $$(0, 0)$$

Question 17

If $$f(x) = x^2$ - 4x + 7$$, for which value of $$x$$ does $$f(x)$$ attain its minimum value?

Accepted Answer

$$x = 2$$

Question 18

Which of the following is the correct difference quotient for $$f(x) = x^2$?

Accepted Answer

$$\frac{f(x+h) - f(x)}{h} = \frac{(x+h)^2$ - x^2$}{h}$$

Graphs and Functions: Rectangular Coordinates, Relations, and Basic Function Concepts

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