2015
Name _______________________________
Pre-Calc w/Trig Semester 1 Exam Review
Chapter 1:
Find the intercepts.
1. 6𝑥 − 3𝑦 = 33
𝑥−4
2. 𝑥 2 + 2𝑦 = 16
3. 𝑦 = 𝑥 2 −7𝑥+10
Test each equation for symmetry with respect to the x-axis, they y-axis, and the origin.
4. 2𝑥 = 3𝑦 2
6. 𝑦 = 𝑥 4 + 2𝑥 2 + 1
5. 𝑥 2 + 4𝑦 2 = 16
7. 𝑥 2 + 𝑥 + 𝑦 2 + 2𝑦 = 0
Write an equation of the line with the given characteristics in slope-intercept and general form.
8. Containing the points (3, -4) and (2, 1)
9. Parallel to the line 2x – 3y = -4 and containing the point (-5, 3)
10. Parallel to the line x + y = 2 and containing the point (1, -3). Write in GENERAL FORM.
11. Perpendicular to the line x + y = 2 and containing the point (4, -3)
12. Perpendicular to the line 3x – y = 4 and containing the point (-2, 4). Write in GENERAL FORM.
13. Find the slope and y-intercept of the linear equation 4x + 6y = 36 .
14. If (-5, -4) is the end of a line segment and (-2, 1) is its midpoint, find the other endpoint.
15. Find the distance d P1,P2  between the points P1 and P2 : P1  4,2 ; P2  3,1

16. Using figure
at the right, Find the equation
of
line in

the
SLOPE-INTERCEPT FORM.

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17. Find the GENERAL FORM of the equation of the circle:
a. with radius = √5 and (h,k) = (-3, -4)
b. with endpoints of a diameter at (5, 9) and (-1, 3)
18. Find the center, radius and intercepts of the circle 𝑥 2 + 𝑦 2 + 24𝑥 − 4𝑦 − 9 = 0
Rational Unit:
Simplify each expression.
h 4
h
h
 2

h  9h  14 h  7h 2  h
2
19.

3
4
2
20.
−
𝑥 2 −𝑦 2
+
𝑦
𝑥+𝑦
21.
1
1

x5 x3
2x 2  3x  5
x 2  2x 15
Solve the equation.

5
x4
2 2
23.
x 1
x x
6
a  49 1
22.
 2

a  7 a  7a a

𝑥
𝑥−𝑦
𝑥 2 +𝑦 2
Chapter 2:
24. Determine if the relation is a function and state the domain and the range.
{(-1, 0), (2, 3), (4, 0)}
25. Given the function 𝑓(𝑥) =
a. f(2)
3𝑥
𝑥 2 −1
b. f(-2)
find each of the following.
c. f(-x)
d. –f(x)
Find the domain of each of the following functions.
26. 𝑓(𝑥) =
𝑥
𝑥 2 −9
27. 𝑓(𝑥) = √2 − 𝑥
𝑓
Find f + g, f – g, f•g, and 𝑔
29. 𝑓(𝑥) = 2 − 𝑥; 𝑔(𝑥) = 3𝑥 + 1
28.
e. f(x – 2)
f. f(2x)
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30. Using the graph of the function shown, find the following.
a.
b.
c.
d.
e.
f.
g.
h.
Find the domain and range.
List the intercepts.
Find f(4).
For what value(s) of x does f(x) = -2?
Increasing/decreasing
Local min/max
Absolute min/max
Is f(1) positive or negative?
Determine algebraically if each of the functions is even, odd, or neither.
31. 𝑓(𝑥) = 𝑥 3 − 4𝑥
4+𝑥 2
33. 𝑓(𝑥) = √1 − 𝑥 3
32. g(𝑥) = 1+𝑥4
34. Which of the following graphs are functions?
35. Know the parent function for the following:
𝑦 = 𝑥2,
𝑦 = 𝑥3,
𝑦 = √𝑥,
3
𝑦 = √𝑥 ,
𝑦=
1
,
𝑥
𝑦 = |𝑥|
Graph each of the following functions using transformations. Describe the transformations.
36. 𝑓(𝑥) = −√𝑥 + 3
37. 𝑔(𝑥) = 3(𝑥 − 1)3 + 1
38. Graph 𝑓(𝑥) = 2(𝑥 + 1)2 − 4 using transformations (shifting, compressing, stretching,
reflecting). Describe the transformations.
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−2𝑥 − 3
39. Given the piecewise function 𝑓(𝑥) = { 1
√𝑥
a.
b.
c.
d.
−4≤𝑥 <0
𝑥=0
𝑥>0
Find the domain and range of the function.
Locate any intercepts.
Graph each function.
Is the function continuous on the domain?
40. Write the equation for the piecewise function in the figure to the right:
41. If (4, -5) is a coordinate on the graph of f(x), identify a coordinate that must be on the graph of the
following:
a) -f(x)
b) f(x + 4)
c) f(x) + 3
d) f(-x)
e) 4f(x)
f) - 3f(x) - 6
42. Suppose the x-intercepts of the graph of y = f(x) are -8 and 1.
a. What are the x-intercepts of y = f(x +4)?
b. What are the x-intercepts of y = f(x - 3)?
c. What are the x-intercepts of y = 2f(x)?
d. What are the x-intercepts of y = f(-x)?
Chapter 3:
43. Determine whether the function is linear or nonlinear:
a.
b.
44. Given 𝑓(𝑥) = 2𝑥 + 4 and 𝑔(𝑥) = −3𝑥 − 6 answer each of the following questions.
a. Solve 𝑓(𝑥) = 0
b. Solve 𝑓(𝑥) > 0
c. Solve 𝑓(𝑥) = 𝑔(𝑥)
d. Solve 𝑓(𝑥) ≤ 𝑔(𝑥)
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45. Given the figure at the right, answer the following questions.
e. For what value of x does 𝑓(𝑥) = 12
f. For what value of x does 𝑓(𝑥) = 0
g. For what interval of x is 𝑓(𝑥) > 6
h. Solve the equation f(x) = g(x).
46. Graph the quadratic function 𝑓(𝑥) = −𝑥 2 − 3𝑥 + 4 by determining whether its graph opens
up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts,
increasing/decreasing, and domain/range.
47. Write the equation 𝑓(𝑥) = −3𝑥 2 + 12𝑥 + 4 in vertex form. State the vertex and the
equation for the axis of symmetry.
48. Write and equation for a parabola in standard form with x-intercepts at -4 and 3 with a vertical stretch
by a factor of -2.
49. If a company’s revenues (R) and costs (C) are represented in the following equations with x
representing the number of units produced, what is the break-even point?
𝑅(𝑥) = 35𝑥
𝐶(𝑥) = 17𝑥 + 75,600
50. Write the equation of a parabola in standard form with a vertex at (4,2) and that passes through the
point (3, 4).
51. Demand Equation The price p, in dollars, and the quantity x sold of a certain product obey
the demand equation.
1
𝑝 = − 6 𝑥 + 100 0 ≤ 𝑥 ≤ 600
a.
b.
c.
d.
e.
Express the revenue R as a function of x (Remember, R = xp).
What is the revenue if 200 units are sold?
Graph the revenue function using a graphing utility.
What quantity of x maximizes revenue? What is the maximum revenue?
What price should the company charge to maximize revenue?
52. Enclosing a Rectangular Field David has available 400 yards of fencing and wishes to
enclose a rectangular area.
a. Express the area A of the rectangle as a function of the width x of the rectangle.
b. What is the domain of A?
c. Graph A = A(x) using a graphing utility. For what value of x is the area largest?
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53. A small manufacturing firm collected the following data on advertising expenditures A (in thousands
of dollars) and total revenue R (in thousands of dollars).
Total
Advertising
Revenue
20
$6,101
22
$6,222
25
$6,350
25
$6,378
27
$6,453
28
$6,423
29
$6,360
31
$6,231
a.
b.
c.
d.
Draw a scatter diagram of the data. What type of relationship exists between the two variables?
Use a graphing utility to find the quadratic function of best fit to these data.
Use the function from b to determine the optimal level of advertising for this firm.
Use the function from b to find the revenue that the firm can expect if it uses the optimal level of
advertising.
54. Solve the inequality algebraically and graphically. Write solution in interval notation.
2x2 + 5x – 12 < 0