Q: Determine the end behavior of $$f(x) = (x - 1)^3$ + 2.$$

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

Question 1

Find the equation of the circle whose diameter has endpoints $$(-4, -2)$$ and $$(2, 6).$$

Accepted Answer

$$(x + 1)^2$ + (y - 2)^2$ = 25$$

Question 2

Find the equation of the line passing through $$(3, -2)$$ that is perpendicular to the line $$3x + 4y = 5.$$

Accepted Answer

$$y = \frac{3}{4}x - \frac{17}{4}$$

Question 3

Given $$f(x) = 2x + 3$$, find and simplify the difference quotient $$\frac{f(x+h) - f(x)}{h}.$$

Accepted Answer

$$2$$

Question 4

Given $$f(x) = x^3$ + 1$$ and $$g(x) = x^2$ - 2x - 6$$, find $$(f \circ g)(2).$$

Accepted Answer

$$-343$$

Question 5

Describe how the graph of $$f(x) = -(x - 2)^2$ + 1$$ can be obtained from the graph of $$y = x^2$.

Accepted Answer

Shift right 2 units, reflect over x-axis, shift up 1 unit

Question 6

Given $$f(x) = \frac{x + 4}{x - 3} is $$one-to-one, find a formula for its inverse function.

Accepted Answer

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

Question 7

Find the zeros of the function $$f(x) = 2x^2$ + 3x + 8.$$

Accepted Answer

$$x = \frac{-3 \pm i\sqrt{55}}{4}$$

Question 8

Find the range and the equation of the axis of symmetry of the quadratic function $$f(x) = -2x^2$ + 10x + 14.$$

Accepted Answer

Range: $$(-\infty, 39]$$, Axis: $$x = 2.5$$

Question 9

Given the quadratic function $$f(x) = -2x^2$ + 8x + 2$$, find the vertex of its graph.

Accepted Answer

$$(2, 10)$$

Question 10

Use synthetic division to find the remainder when $$x^3$$ + $$3x^2 - 10x$ is divided by x - 2$.

Accepted Answer

$$6$$

Question 11

Use the remainder theorem and synthetic division to find $$f(-2)$$ given $$f(x) = x^4$ + 2x^3$ - 10x + 1.$$

Accepted Answer

$$-39$$

Question 12

Determine the end behavior of $$f(x) = (x - 1)^3$ + 2.$$

Accepted Answer

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

Question 13

Given $$f(x) = x^3$ + 1$$ and $$g(x) = x^3$ - 1$$, find $$(g \circ f)(x).$$

Accepted Answer

$$(x^3$$ + $$1)^3 - 1$

Question 14

Which of the following functions has an inverse that is also a function?

Accepted Answer

$$f(x) = x^3$ - 6$$

Question 15

Approximate $$\log_{0.3}(100) to 4 $$decimal places.

Accepted Answer

$$-4.1918$$

Question 16

A wood sculptor carves Totem Poles, Bears, and Deers. Each Totem Pole requires 2 hours carving, 1 hour sanding, 3 hours painting; each Bear requires 2 hours carving, 2 hours sanding, 2 hours painting; each Deer requires 1 hour carving, 2 hours sanding, 2 hours painting. If the sculptor spends 14 hours carving, 15 hours sanding, and 21 hours painting, how many of each statue were made?

Accepted Answer

Totem Poles: 2, Bears: 3, Deers: 4

Question 17

Expand $$\log_3$ \left( \frac{4x^3$}{8y^2$} \right) as $$much as possible.

Accepted Answer

$$\log_3$ 4 + 3\log_3$ x - \log_3$ 8 - 2\log_3$ y$$

Question 18

Solve $$\log_2$ (x + 20) + \log_2$ (x - 3) = 2.$$

Accepted Answer

$$x = 5$$

Question 19

Solve $$e^{2x + 5} = 40$. Approximate your answer to 4 decimal places.

Accepted Answer

$$x = -1.0471$$

Question 20

In 2011, the population of Texas was 25.67 million with a growth rate of 2.06% per year. Estimate the population size in 2022 using the model $$P(t) = P_0$ e$$^{kt}$$.

Accepted Answer

$$31.97$$ million

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