Question 1

Which of the following is the correct vertex form of a quadratic function, and what does each parameter represent?

Accepted Answer

$$f(x) = a(x - h)^2$ + k$$, where $$(h, k) is $$the vertex and $$a$$ determines the direction and width of the parabola.

Question 2

Given the quadratic function $$f(x) = -2(x + 3)^2$ + 5$$, which statement about its graph is true?

Accepted Answer

The parabola opens downward and has a maximum at $$(-3, 5).$$

Question 3

For the quadratic function $$f(x) = 4x^2$ - 8x + 7$$, what are the coordinates of the vertex?

Accepted Answer

$$(1, 3)$$

Question 4

A ball is thrown and its height in feet is modeled by $$f(x) = -0.2x^2$ + 1.7x + 5$$, where $$x is $$the horizontal distance in feet. What is the maximum height the ball reaches, and at what horizontal distance does this occur? (Round to the nearest tenth.)

Accepted Answer

Maximum height is $$6.4$$ feet at $$4.3$$ feet from the throw point.

Question 5

Given $$f(x) = x^2$ - 4x - 12$$, what is the equation of the axis of symmetry?

Accepted Answer

$$x = 2$$

Question 6

A food truck vendor has weekly fixed costs of $$540$$ dollars and variable costs of $$5.45$$ dollars per gyro combo. If $$x is $$the number of combos sold, what is the cost function $$C(x)$$?

Accepted Answer

$$C(x) = 5.45x + 540$$

Question 7

For the quadratic function $$f(x) = -3x^2$ + 30x - 3$$, determine whether the function has a maximum or minimum value, and find the $$x-$$coordinate where it occurs.

Accepted Answer

Maximum at $$x = 5$$

Question 8

A rectangular plot of land borders a river and is to be fenced on only three sides (not along the river). If $$250$$ feet of fencing is available, what dimensions maximize the area, and what is the maximum area?

Accepted Answer

Width $$= 62.5 ft$$, Length $$= 125 ft$$, Area $$= 7812.5 sq ft$$

Question 9

Given the quadratic function $$f(x) = 4(x - 3)^2$ + 2$$, what is the vertex of the parabola?

Accepted Answer

$$(3, 2)$$

Question 10

For the quadratic function $$f(x) = x^2$ + 6x + 3$$, what is the $$y-$$intercept?

Accepted Answer

$$(0, 3)$$

Question 11

A shot put is thrown and its height is modeled by $$f(x) = -0.05x^2$ + 2.7x + 5.4. $$What is the initial height from which the shot is released?

Accepted Answer

$$5.4$$ feet

Question 12

Given the quadratic function $$f(x) = 6x^2$ + 12x - 7$$, what is the equation of the axis of symmetry?

Accepted Answer

$$x = -1$$

Question 13

For the quadratic function $$f(x) = 3x^2$ - 30x - 4$$, what is the range of the function?

Accepted Answer

$$[ -\infty, -76 ]$$

Question 14

If the vertex of a parabola is $$(-5, -8)$$ and the parabola opens downward, what is the range of the function?

Accepted Answer

$$(-\infty, -8]$$

Question 15

A quadratic function has a minimum at $$x = -2$$ and the minimum value is $$5. $$What is the range of the function?

Accepted Answer

$$[5, \infty )$$

Question 16

Given the quadratic function $$f(x) = (x - 3)^2$ - 4$$, what is the domain and range of the function?

Accepted Answer

Domain: $$(-\infty, \infty)$$, Range: $$[ -4, \infty )$$

Question 17

The profit function for a food truck is $$P(x) = -0.001x^2$ + 3.28x - 540$$, where $$x is $$the number of gyro combos sold. How many combos should be sold to maximize profit, and what is the maximum profit? (Round to the nearest integer for $$x$$ and two decimals for profit.)

Accepted Answer

$$x = 1640$$, Maximum profit $$= \ 2752.80$

Question 18

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

Accepted Answer

If $$a \u003e 0$$, the parabola opens upward and the vertex is a minimum.

Question 19

Given $$f(x) = 4(x - 1)^2$ - 3$$, what is the axis of symmetry?

Accepted Answer

$$x = 1$$

Question 20

For the quadratic function $$f(x) = 4x^2$ - x + 12$$, what is the domain and range?

Accepted Answer

Domain: $$(-\infty, \infty)$$, Range: $$[11.984, \infty)$$

Step-by-Step Guidance for Quadratic Functions in College Algebra

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