Q: Which of the following best describes the end behavior of $$f(x) = 2x^2$ - 12x - 9$$?

As $$x \to \infty$$, $$f(x) \to \infty$$; as $$x \to -\infty$$, $$f(x) \to \infty.$$

Question 1

Given the function $$f(x) = 2x^2$ - 12x - 9$$, which of the following statements is true about its minimum or maximum value?

Accepted Answer

The function has a minimum value because the leading coefficient is positive.

Question 2

For the quadratic function $$f(x) = 2x^2$ - 12x - 9$$, at which $$x-$$value does the minimum value occur?

Accepted Answer

$$x = 3$$

Question 3

What is the minimum value of $$f(x) = 2x^2$ - 12x - 9$$?

Accepted Answer

$$-27$$

Question 4

What is the domain of the function $$f(x) = 2x^2$ - 12x - 9$$?

Accepted Answer

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

Question 5

What is the range of the function $$f(x) = 2x^2$ - 12x - 9$$?

Accepted Answer

$$[-27, \infty)$$

Question 6

Which of the following best describes the vertex of the parabola $$f(x) = 2x^2$ - 12x - 9$$?

Accepted Answer

It is a minimum point at $$(3, -27).$$

Question 7

A company models its profit with $$P(x) = 2x^2$ - 12x - 9$$, where $$x is $$the number of units produced. What is the minimum profit, and at how many units does it occur?

Accepted Answer

Minimum profit is $$-27 at$$ $$x = 3$$ units.

Question 8

If $$f(x) = 2x^2$ - 12x - 9$$ models the height of a projectile, what does the minimum value represent in a real-world context?

Accepted Answer

The lowest point the projectile reaches.

Question 9

Which of the following is the correct method to find the vertex of $$f(x) = 2x^2$ - 12x - 9$$?

Accepted Answer

Use $$x = -\frac{b}{2a}$$ where $$a = 2$$ and $$b = -12.$$

Question 10

If the function $$f(x) = 2x^2$ - 12x - 9 is $$shifted up by 10 units, what is the new minimum value?

Accepted Answer

$$-17$$

Question 11

Suppose $$f(x) = 2x^2$ - 12x - 9$$ represents the cost in dollars to produce $$x$$ items. If the company wants to minimize costs, how many items should they produce?

Accepted Answer

$$3$$

Question 12

Which transformation would change $$f(x) = 2x^2$ - 12x - 9 so $$that its minimum occurs at $$x = 5$$?

Accepted Answer

Replace $$x$$ with $$x - 2 in $$the function.

Question 13

If $$f(x) = 2x^2$ - 12x - 9 is $$rewritten in vertex form, what is the equation?

Accepted Answer

$$f(x) = 2(x - 3)^2$ - 27$$

Question 14

Which of the following intervals best describes the set of $$x-$$values for which $$f(x) = 2x^2$ - 12x - 9 \leq 0$$?

Accepted Answer

$$x \in [-1, 6]$$

Question 15

If $$f(x) = 2x^2$ - 12x - 9$$ models the temperature in degrees Celsius at time $$x$$ hours, what is the lowest temperature and when does it occur?

Accepted Answer

Lowest temperature is $$-27 at$$ $$x = 3$$ hours.

Question 16

Which of the following best describes the end behavior of $$f(x) = 2x^2$ - 12x - 9$$?

Accepted Answer

As $$x 	o \infty$$, $$f(x) 	o \infty$$; as $$x 	o -\infty$$, $$f(x) 	o \infty.$$

Question 17

If $$f(x) = 2x^2$ - 12x - 9 is $$reflected over the $$x-$$axis, what is the new minimum or maximum value?

Accepted Answer

Maximum value is $$27 at$$ $$x = 3.$$

Question 18

Which of the following is the axis of symmetry for $$f(x) = 2x^2$ - 12x - 9$$?

Accepted Answer

$$x = 3$$

Question 19

If $$f(x) = 2x^2$ - 12x - 9$$ represents the cost to produce $$x$$ items, and the company must produce at least 4 items, what is the minimum cost?

Accepted Answer

$$f(4) = 2(4)^2$ - 12(4) - 9 = 32 - 48 - 9 = -25$$

Quadratic Functions: Minimum, Maximum, Domain, and Range

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