Question 1

Which of the following best describes the graph of any quadratic function $$f(x) = ax^2$ + bx + c$$ where $$a 
eq 0$$?

Accepted Answer

A parabola

Question 2

What is the domain of any quadratic function $$f(x) = ax^2$ + bx + c$$ where $$a 
eq 0$$?

Accepted Answer

$$x \in \mathbb{R}$$

Question 3

Given the quadratic function $$f(x) = -x^2$ + 5x + 3$$, what is the axis of symmetry?

Accepted Answer

$$x = 2.5$$

Question 4

For the quadratic function $$f(x) = 2x^2$ - 5x - 5$$, what is the $$y-$$intercept?

Accepted Answer

$$y = -5$$

Question 5

Which formula is used to find the $$x-$$intercepts (real zeros) of a quadratic function $$ax^2 + bx + c = 0$?

Accepted Answer

$$x = \frac{-b \pm \sqrt{b^2$ - 4ac}}{2a}$$

Question 6

A company’s profit from selling $$x$$ units is given by $$P(x) = -4x^2$ + 200x - 160. $$How many units should be sold to maximize profit?

Accepted Answer

$$x = 25$$

Question 7

Given the quadratic function $$f(x) = ax^2$ + bx + c$$, what is the $$x-$$coordinate of the vertex?

Accepted Answer

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

Question 8

If the cost to build a shed is given by $$C(x) = 0.01x^2$ - 20x + 25000$$, where $$x is $$the area in square feet, what is the minimum cost and for what value of $$x$$ does it occur?

Accepted Answer

Minimum at $$x = 1000$$, $$C(1000) = 24000$$

Question 9

A company’s revenue and cost functions are $$R(x) = 300x - x^2$ and C(x) = 40x + 1600$. What is the break-even point (to the nearest 1000 units)?

Accepted Answer

$$x = 4$$ and $$x = 296$$

Question 10

For the quadratic function $$f(x) = -2x^2$ + 8x - 3$$, which statement is true about its graph?

Accepted Answer

It opens downward and has a maximum value.

Question 11

If a quadratic function models the cost $$C(x) of $$producing $$x$$ items and $$C(x)$$ has a minimum at $$x = 500$$, what does this mean in a real-world context?

Accepted Answer

The cost is lowest when 500 items are produced.

Question 12

Given $$f(x) = 2x^2$ - 5x - 5$$, what are the $$x-$$intercepts to four decimal places?

Accepted Answer

$$x = -1.0$$ and $$x = 2.5$$

Question 13

A company’s cost function is $$C(x) = 0.04x^2$ - 20x + 6000$$, where $$x is $$the number of hours employees work. What number of hours minimizes the cost?

Accepted Answer

$$x = 250$$

Question 14

Which of the following statements about the vertex of a quadratic function $$f(x) = a(x-h)^2$ + k is $$correct?

Accepted Answer

The vertex is at $$(h, k)$$ and is a maximum if $$a \u003c 0.$$

Question 15

If the quadratic equation $$ax^2 + bx + c = 0$ has b^2$ - 4ac \u003c 0$$, what does this mean about its real $$x-$$intercepts?

Accepted Answer

There are no real $$x-$$intercepts.

Question 16

A company’s profit function is $$P(x) = -2x^2$ + 40x - 160. $$What is the maximum profit?

Accepted Answer

$$240$$

Question 17

Which of the following intervals describes where the function $$f(x) = x^2$ - 4x + 3 is $$decreasing?

Accepted Answer

$$(-\infty, 2)$$

Question 18

If a quadratic function $$f(x) = ax^2$ + bx + c$$ has $$a \u003e 0$$, which of the following is true?

Accepted Answer

The parabola opens upward and the vertex is a minimum.

Question 19

A company’s cost to produce $$x$$ units is $$C(x) = 0.01x^2$ - 20x + 25000. $$What is the cost to produce 1500 units?

Accepted Answer

$$C(1500) = 22500$$

2.3 Quadratic Functions: Properties, Graphs, and Applications

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