Question 1

Given the vertex $$(-2, 5)$$ and a point $$(0, -7) on a $$parabola, what is the equation of the quadratic function in vertex form?

Accepted Answer

$$y = -3(x + 2)^2$ + 5$$

Question 2

If $$a \u003e 0 in $$the vertex form $$y = a(x - h)^2$ + k$$, what is the minimum value of the function?

Accepted Answer

$$k at$$ $$x = h$$

Question 3

A projectile is launched from a cliff 500 feet above the ground at an angle of $$45^	ext{o}$$ with a velocity of 100 ft/s. Its height $$h$$ after $$t$$ seconds is given by $$h = -16t^2$ + 100t + 500. $$How long does it take for the projectile to hit the water?

Accepted Answer

$$t \approx 43.36$$ seconds

Question 4

For the projectile in the previous question, when the height is 1000 feet, how far is it from the base of the cliff?

Accepted Answer

$$x \approx 447.36 ft$

Question 5

A farmer has 2000 yards of fence to enclose a rectangular field. What is the maximum area that can be enclosed?

Accepted Answer

$$250,000$$ square yards

Question 6

Given the fencing problem, what should the dimensions of the rectangle be to maximize the area?

Accepted Answer

$$w = 500$$ yards, $$l = 500$$ yards

Question 7

If the price $$p$$ and quantity $$x$$ sold are related by $$p = \frac{500 - x}{100}$$, what is the revenue $$R as a $$function of $$x$$?

Accepted Answer

$$R = x \cdot \frac{500 - x}{100}$$

Question 8

For the revenue function $$R = x \cdot \frac{500 - x}{100}$$, what quantity $$x$$ maximizes the revenue?

Accepted Answer

$$x = 250$$

Question 9

What is the maximum revenue for the function $$R = x \cdot \frac{500 - x}{100}$$?

Accepted Answer

$$625$$

Question 10

For the revenue function above, what price should the company charge to maximize the revenue?

Accepted Answer

$$p = 2.5$$

Question 11

Which of the following best describes the effect of the parameter $$a in $$the vertex form $$y = a(x - h)^2$ + k$$?

Accepted Answer

It determines the direction and width of the parabola.

Question 12

In the context of optimization, what does the vertex of a quadratic function represent?

Accepted Answer

The maximum or minimum value of the function.

Question 13

A rectangular field is to be enclosed with 2000 yards of fencing. If the width is $$w$$ yards, what is the length $$l in $$terms of $$w$$?

Accepted Answer

$$l = 2000 - 2w$$

Question 14

If a quadratic function opens downward, what can be said about the coefficient $$a in$$ $$y = a(x - h)^2$ + k$$?

Accepted Answer

$$a \u003c 0$$

Question 15

A company finds that the demand for its product is $$p = \frac{500 - x}{100}. $$What is the domain of the revenue function $$R(x)$$?

Accepted Answer

$$0 \leq x \leq 500$$

Question 16

Which of the following is NOT a step in solving a quadratic optimization problem in a real-world context?

Accepted Answer

Ignore constraints and maximize the function over all real numbers.

Question 17

If the vertex of a parabola is at $$(h, k)$$ and $$a \u003c 0$$, what is the maximum value of the function?

Accepted Answer

$$k$$

Question 18

A projectile is launched from a cliff. Its height is modeled by $$h = -16t^2$ + vt + h_0$. What does the -16$ represent?

Accepted Answer

The acceleration due to gravity (in ft/s$$^2)$$

Question 19

In the fencing problem, if the area is $$A = w(1000 - w)$$, what value of $$w$$ maximizes the area?

Accepted Answer

$$w = 500$$

Quadratic Functions: Vertex Form, Optimization, and Applications (Precalculus Sections 3.3–3.4)

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