Q: The weight $$W of an $$object varies inversely as the square of the distance $$d$$ from the center of the Earth. At sea level ($$d = 3978$$ miles), an astronaut weighs $$220$$ pounds. What is the astronaut's weight when he is $$200$$ miles above the surface?

$$W = 220 \times \left(\frac{3978}{4178}\$$right)^2$$ \approx 199$$ pounds

Q: Find the domain and all asymptotes of the function $$f(x) = \frac{x^2$ + 3x + 1}{4x^2$ - 9}.$$

Domain: $$x \neq \frac{3}{2}, x \neq -\frac{3}{2}$$; Vertical asymptotes: $$x = \frac{3}{2}, x = -\frac{3}{2}$$

Question 1

Find the $$x- $$and $$y-$$intercepts of the line $$5x - 2y = 10$$ and graph the line. What are the intercepts?

Accepted Answer

$$x-$$intercept: $$(2, 0)$$; $$y-$$intercept: $$(0, -5)$$

Question 2

Given the function $$f(x) = 2x^2$ - x + 5$$, evaluate $$f(-1).$$

Accepted Answer

$$f(-1) = 8$$

Question 3

Find the equation of the line containing the points $$(4, -10)$$ and $$(-8, -12). $$What is the slope?

Accepted Answer

Slope $$m = \frac{-2}{12} = -\frac{1}{6}$$

Question 4

Find the equation of the line parallel to $$2x + 3y = -6$$ and passing through the point $$(-1, 3).$$

Accepted Answer

$$2x + 3y = 7$$

Question 5

The weight $$W of an $$object varies inversely as the square of the distance $$d$$ from the center of the Earth. At sea level ($$d = 3978$$ miles), an astronaut weighs $$220$$ pounds. What is the astronaut's weight when he is $$200$$ miles above the surface?

Accepted Answer

$$W = 220 	imes \left(\frac{3978}{4178}\$$right)^2$$ \approx 199$$ pounds

Question 6

Find the equation of the line perpendicular to $$2x + 3y = -6$$ and passing through the point $$(-1, 3).$$

Accepted Answer

$$3x - 2y = -9$$

Question 7

Solve the inequality $$2x - 1 \u003c x - 3$$ and express the solution in interval notation.

Accepted Answer

$$x \u003c -2$$

Question 8

A boat moves at a rate of $$11 mi/h in $$still water. If the river current flows at a rate of $$5 mi/h$$, how long does it take for the boat to travel $$33$$ miles upstream?

Accepted Answer

$$\frac{33}{6}$$ hours

Question 9

Miguel deposits $$8,000$$ into two accounts: a CD that offers $$4\%$$ interest rate and a Mutual Bond with $$5\%$$ interest rate. If the total interest earned after a year is $$380$$, how much was deposited into each account?

Accepted Answer

$$x = 3,000 in CD$$, $$y = 5,000 in $$Bond

Question 10

Given $$m(x) = x^3$ - 5x$$, find and simplify the difference quotient $$\frac{m(x+h) - m(x)}{h}.$$

Accepted Answer

$$3x^2h$$ + $$3xh^2$$ + $$h^3 - 5h$

Question 11

Given the quadratic equation $$y = 5x^2$ - 10x + 7$$, what is the axis of symmetry?

Accepted Answer

$$x = 1$$

Question 12

Find the domain and all asymptotes of the function $$f(x) = \frac{x^2$ + 3x + 1}{4x^2$ - 9}.$$

Accepted Answer

Domain: $$x 
eq \frac{3}{2}, x 
eq -\frac{3}{2}$$; Vertical asymptotes: $$x = \frac{3}{2}, x = -\frac{3}{2}$$

Question 13

While on top of a $$24-ft $$building, an object is thrown into the air. The height of the object is given by $$h(t) = -16t^2$ + 24t + 24$$, where $$t$$ denotes the time (in seconds) after the ball was thrown. What is the maximum height of the ball and when does it occur?

Accepted Answer

Maximum height: $$42 ft at$$ $$t = 0.75 sec$$

Question 14

Given the point $$(1, 5) on $$the graph of $$y = f(x)$$, find the corresponding point for the graph of $$y = f(x - 2) + 3.$$

Accepted Answer

$$(3, 8)$$

Question 15

Simplify $$(3 + 4i)(4 - 3i)$$ and express your answer in standard form $$(a + bi).$$

Accepted Answer

$$24 + 7i$$

Question 16

Find the domain of $$h(y) = \frac{9}{y - 4}.$$

Accepted Answer

All real $$y$$ except $$y = 4$$

Question 17

A parabola has a vertex at the point $$(3, 4)$$ and passes through the point $$(-1, 7). $$Write its equation in vertex form.

Accepted Answer

$$y = \frac{1}{4}(x - 3)^2$ + 4$$

Question 18

If the point $$(7, -1) is on $$the graph of $$y = f(x)$$, what is the corresponding point on the graph of $$y = -2f(x - 3) + 5$$?

Accepted Answer

$$(10, 7)$$

Question 19

Turnip yield $$T is $$related to job data inversely as the number of people $$P$$ working on that job. If it takes $$5$$ hours for $$7$$ bricklayers to build a wall, how long will it take $$10$$ bricklayers to complete the same wall?

Accepted Answer

$$3.5$$ hours

Precalculus Midterm Exam Review: Key Concepts and Practice Problems

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