Question 1

Given the quadratic function $$f(x) = 2(x - 3)^2$ - 5$$, what are the coordinates of the vertex?

Accepted Answer

$$(3, -5)$$

Question 2

For the quadratic function $$f(x) = -x^2$ + 6x - 8$$, determine whether the parabola opens upward or downward and find the axis of symmetry.

Accepted Answer

Opens downward; axis of symmetry $$x = 3$$

Question 3

A company models its profit with the function $$P(x) = -4x^2$ + 40x + 100$$, where $$x is $$the number of units sold (in thousands). What is the maximum profit, and how many units should be sold to achieve it?

Accepted Answer

Maximum profit is $$300 at$$ $$x = 5$$ thousand units

Question 4

Which of the following is the correct formula for the $$x-$$coordinate of the vertex of a quadratic function $$f(x) = ax^2$ + bx + c$$?

Accepted Answer

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

Question 5

A rectangular garden is to be fenced on all four sides using 80 meters of fencing. What dimensions will maximize the area of the garden, and what is the maximum area?

Accepted Answer

Dimensions: $$20 m by$$ $$20 m$$; Maximum area: $$400 m$^2$$

Question 6

Given $$f(x) = -2(x + 4)^2$ + 7$$, what is the range of $$f$$?

Accepted Answer

$$(-\infty, 7]$$

Question 7

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

Accepted Answer

The parabola opens upward and has a minimum value.

Question 8

A ball is thrown upward from a height of 2 meters with an initial velocity of 10 m/s. Its height after $$t$$ seconds is given by $$h(t) = -5t^2$ + 10t + 2. $$What is the maximum height reached by the ball?

Accepted Answer

$$7$$ meters

Question 9

Which of the following is the $$y-$$intercept of the quadratic function $$f(x) = 3x^2$ - 5x + 2$$?

Accepted Answer

$$2$$

Question 10

Given $$f(x) = (x - 2)^2$ - 9$$, for which values of $$x$$ does $$f(x) = 0$$?

Accepted Answer

$$x = 5$$ and $$x = -1$$

Question 11

A farmer wants to build a rectangular pen against a barn using 100 meters of fencing for the other three sides. What dimensions maximize the area, and what is the maximum area?

Accepted Answer

Width: $$25 m$$, Length: $$50 m$$; Maximum area: $$1250 m$^2$$

Question 12

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

Accepted Answer

The vertex is at $$(h, k)$$ and the axis of symmetry is $$x = h.$$

Question 13

If the quadratic function $$f(x) = x^2$ - 8x + 20$$ models the cost (in dollars) to produce $$x$$ items, what is the minimum cost and at how many items is it achieved?

Accepted Answer

Minimum cost is $$4$$ dollars at $$x = 4$$ items

Question 14

Which of the following best describes the domain and range of $$f(x) = -3x^2$ + 12x - 7$$?

Accepted Answer

Domain: $$(-\infty, \infty)$$; Range: $$(-\infty, 5]$$

Question 15

A quadratic function has its vertex at $$(4, -2)$$ and opens upward. Which of the following could be its equation?

Accepted Answer

$$f(x) = a(x - 4)^2$ - 2$$, where $$a \u003e 0$$

Question 16

If $$f(x) = 5 - (x + 1)^2$, what is the maximum value of f(x$$)$$ and at what x$ does it occur?

Accepted Answer

Maximum value is $$5 at$$ $$x = -1$$

Question 17

A quadratic function $$f(x) = ax^2$ + bx + c$$ has $$a \u003c 0. $$Which of the following is true about its graph?

Accepted Answer

It opens downward and has a maximum value.

Question 18

Given $$f(x) = -x^2$ + 4x + 1$$, what are the $$x-$$intercepts of the function?

Accepted Answer

$$x \approx -0.2$$ and $$x \approx 4.2$$

Question 19

A rectangular playground is to be built using 200 meters of fencing. What is the largest possible area of the playground?

Accepted Answer

$$2500 m$^2$$

Quadratic Functions: Properties, Graphing, and Applications

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