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

There are no real $x$-intercepts.

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 $a < 0$.

Question 15

If the quadratic equation $ax^2 + bx + c = 0$ has $b^2 - 4ac < 0$, 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 $a > 0$, 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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