Q: Use the Leading Coefficient Test to determine the end behavior of $$f(x) = -x^5$ + 4x^3$ - 5x. $$Which statement is correct?

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

Question 1

Solve the following system of equations: 
$$\begin{cases} 4x + 2y = 3 \ 3x - y = -4 \end{cases}$$
What is the solution?

Accepted Answer

$$(2, 3)$$

Question 2

Given the points $$(1, -4)$$ and $$(5, 2)$$, what is the distance between them? Express your answer in simplified radical form.

Accepted Answer

$$2\sqrt{13}$$

Question 3

Given the equation $$x^2 + 6x$$ + $$y^2 + 2y = 1$, write it in standard form by completing the square. What is the center of the circle?

Accepted Answer

$$(3, -1)$$

Question 4

For the quadratic function $$f(x) = 3(x+1)^2$ - 12$$, what is the vertex?

Accepted Answer

$$(-1, -12)$$

Question 5

Given $$f(x) = 3(x+1)^2$ - 12$$, what is the axis of symmetry?

Accepted Answer

$$x = -1$$

Question 6

A rain gutter is made from sheets of aluminum that are 18 inches wide by turning up the edges to form right angles. What depth will maximize the cross-sectional area?

Accepted Answer

$$4.5$$ inches

Question 7

Find the domain of the rational function $$f(x) = \frac{x^3$ + 2}{x^2$ - 16}.$$

Accepted Answer

$$(-\infty, -4) \cup (-4, 4) \cup (4, \infty)$$

Question 8

For the polynomial $$f(x) = (x+1)(x-2)(x-1)^2$, what is the multiplicity of the zero at x=1$ and does the graph touch or cross the x-axis at this point?

Accepted Answer

Multiplicity 2, touches and turns around

Question 9

Use the Leading Coefficient Test to determine the end behavior of $$f(x) = -x^5$ + 4x^3$ - 5x. $$Which statement is correct?

Accepted Answer

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

Question 10

Given the rational function $$f(x) = \frac{x^2$ - 1}{x(x+1)}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 0$$ and $$x = -1$$

Question 11

A drug's concentration is modeled by $$C(t) = \frac{5t^2$}{2t^2$ + 2}. $$What is the horizontal asymptote, and what does it mean in practical terms?

Accepted Answer

Horizontal asymptote $$y = 0$$; the drug concentration approaches zero as time increases.

Question 12

If $$y$$ varies directly as $$z$$ and $$y = 160$$ when $$z = 16$$, what is $$y$$ when $$z = 15$$?

Accepted Answer

$$150$$

Question 13

The intensity $$I of $$light varies inversely as the square of the distance $$D$$ from the source. If $$I = 2.5$$ foot-candles at $$D = 28 ft$$, what is the intensity at $$D = 40 ft$$?

Accepted Answer

$$1.225$$ foot-candles

Question 14

Body-mass index (BMI) varies directly as weight and inversely as the square of height. If a person who weighs 170 pounds and is 71 inches tall has a BMI of 23.71, what is the BMI for a person who weighs 122 pounds and is 66 inches tall (to the nearest tenth)?

Accepted Answer

$$19.7$$

Question 15

Which of the following is the equation of a circle with center $$(5, -3)$$ and radius $$5$$?

Accepted Answer

$$(x-5)^2$$ + $$(y+3)^2 = 25$

Question 16

Given the function $$f(x) = -x^2$ - 2x - 9$$, does it have a minimum or maximum value, and what are the coordinates of that point?

Accepted Answer

Maximum at $$(-1, -8)$$

Question 17

Write the equation of a polynomial $$P(x)$$ with the following properties: touches the x-axis at $$x=0$$, crosses the x-axis at $$x=2$$, and lies below the x-axis between $$x=0$$ and $$x=2. $$Use the smallest possible degree and a leading coefficient of $$1 or$$ $$-1.$$

Accepted Answer

$$P(x) = x^2$(x-2)$$

Question 18

For the rational function $$f(x) = \frac{4x-3}{5x+3}$$, what is the horizontal asymptote?

Accepted Answer

$$y = \frac{4}{5}$$

Math 120 – Unit 3 Exam Review: College Algebra Study Guidance

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