Q: A quadratic function models the height $$h (in $$meters) of a projectile at time $$t (in $$seconds): $$h(t) = -4.9t^2$ + 20t + 2. $$What is the maximum height reached by the projectile?

Find $$t at $$vertex: $$t = \frac{-20}{2 \times -4.9} = 2.04$$, then $$h(2.04) = -4.9 \times (2.04)^2$ + 20 \times 2.04 + 2 \approx 22.4$$ meters

Q: If $$f(x) = x^2$ + 4x + 5$$, what is the y-coordinate of the vertex?

First, $$x = \frac{-4}{2 \times 1} = -2$$, then $$y = (-2)^2$ + 4 \times (-2) + 5 = 4 - 8 + 5 = 1$$

Question 1

Which of the following statements best describes the Vertical Line Test for functions?

Accepted Answer

A graph passes the Vertical Line Test if every vertical line touches the graph at exactly one point.

Question 2

Given the quadratic function $$f(x) = 2x^2$ - 4x + 1$$, what is the x-coordinate of its vertex?

Accepted Answer

$$x = \frac{-b}{2a} = \frac{4}{4} = 1$$

Question 3

A rational function $$f(x) = \frac{3x^2$ + 2}{x^2$ - 5} is $$given. What is the horizontal asymptote?

Accepted Answer

$$y = 3$$

Question 4

Which of the following is NOT a function?

Accepted Answer

A graph where a vertical line touches the graph at more than one point.

Question 5

For the quadratic function $$f(x) = -x^2$ + 6x - 8$$, what is the direction in which the parabola opens?

Accepted Answer

Downward, because the leading coefficient is negative.

Question 6

A rational function $$f(x) = \frac{2x^3$ + 1}{x^2$ - 4} is $$given. What is the horizontal asymptote?

Accepted Answer

There is no horizontal asymptote because the degree of the numerator is greater than the denominator.

Question 7

Which transformation affects the horizontal movement of a function $$f(x)$$?

Accepted Answer

Adding a constant inside the function, $$f(x + c)$$

Question 8

Given $$f(x) = \frac{5x^2$ - 3x + 2}{2x^2$ + 7}$$, what is the horizontal asymptote?

Accepted Answer

$$y = \frac{5}{2}$$

Question 9

A quadratic function models the height $$h (in $$meters) of a projectile at time $$t (in $$seconds): $$h(t) = -4.9t^2$ + 20t + 2. $$What is the maximum height reached by the projectile?

Accepted Answer

Find $$t at $$vertex: $$t = \frac{-20}{2 	imes -4.9} = 2.04$$, then $$h(2.04) = -4.9 	imes (2.04)^2$ + 20 	imes 2.04 + 2 \approx 22.4$$ meters

Question 10

If $$f(x) = x^2$ + 4x + 5$$, what is the y-coordinate of the vertex?

Accepted Answer

First, $$x = \frac{-4}{2 	imes 1} = -2$$, then $$y = (-2)^2$ + 4 	imes (-2) + 5 = 4 - 8 + 5 = 1$$

Question 11

Which of the following rational functions has a horizontal asymptote at $$y = 0$$?

Accepted Answer

$$f(x) = \frac{2x}{x^2$ + 1}$$

Question 12

A function $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 1}{x^2$ + 1}. $$What is the range of $$f(x)$$?

Accepted Answer

$$(-1, 1)$$

Question 13

Which scenario best illustrates the function rule in a real-world context?

Accepted Answer

Each patient in a hospital is assigned to exactly one room.

Question 14

For the rational function $$f(x) = \frac{4x^2$ + 3x + 1}{2x^2$ - 5x + 6}$$, what is the horizontal asymptote?

Accepted Answer

$$y = \frac{4}{2} = 2$$

Question 15

Which of the following transformations will reflect the graph of $$f(x)$$ downward?

Accepted Answer

Multiply $$f(x) by$$ $$-1$$: $$-f(x)$$

Question 16

A rational function $$f(x) = \frac{2x^2$ + 3x + 1}{x^2$ - 4x + 4} is $$given. What is the horizontal asymptote?

Accepted Answer

$$y = 2$$

Question 17

If the degree of the numerator of a rational function is less than the degree of the denominator, what is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 18

Which of the following best describes the effect of a positive leading coefficient in a quadratic function?

Accepted Answer

The parabola opens upward.

Question 19

A rational function $$f(x) = \frac{3x^2$ + 2x + 1}{x^2$ - x - 6} is $$given. What is the horizontal asymptote?

Accepted Answer

$$y = 3$$

Pre-Calculus Toolkit: Function Families, Quadratics, and Rational Functions

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