Math 111 Chapter 4
Non-linear Functions & Equations Test
Winter2016/CHutt
Name ______Key__________________
Directions: This is an open book test. You will have to use a graphing calculator to complete this test. For
all graphs, plot points to make your graph more accurate. You will be graded on accuracy.
1. Graph the function 𝑃(𝑥) = −3𝑥 4 + 2𝑥 3 + 20𝑥 2 − 10𝑥 − 25 below. (20 pts)
a. Label all local extrema with correct coordinates rounded to the tenths.
b. Give the intervals of increase for this function:
(−∞, −𝟏. 𝟕) ∪ (𝟎. 𝟐, 𝟐. 𝟎)
c. The rational root theorem gives 12 possible roots (including signs). What are they?
𝟏
𝟓
𝟐𝟓
𝟑
𝟑
𝟑
±𝟏, ±𝟓, ±𝟐𝟓, ± , ± , ±
_______________________________
d. Write 𝑃(𝑥) in completely factored form. Justify your work using synthetic division and/or other
algebraic methods.
5
I see the roots 𝑥 = −1 and 𝑥 = .
3
2. ℎ(𝑥) = −7 − 8𝑥 2 − 𝑥 3 + 2𝑥 4 + 3𝑥 5 . Do not graph this function, that is a a waste of time. (10 pts)
a) What is the degree ofℎ(𝑥)? ___5___
b) What is the leading coefficient ofℎ(𝑥)? ____3____
c) Which graph is the only one which might be ℎ(𝑥)? __d_____
a
c
b
d
d) Divide to find ℎ(2) and show your work.
2
3
3
2
6
8
-1
16
15
-8 0 -7
30 44 88
22 44 81
3. Write “even,” “odd,” or “neither” below to describe each function representation given. (10 pts)
b) 𝑃(𝑥) = 𝑥 6 − 3𝑥 4 + 20𝑥 2 − 25
a)
____neither_______
c)
______even____
d)
x
y
______odd__________
-4
5
-3
2
-2
1
-1
0
0
1
____neither________
1
2
2
5
3
8
5. Solve the following equations and inequalities. (20 points)
a.
√3𝑥 + 1 = 𝑥 − 3
3𝑥 + 1 = (𝑥 − 3)2
b.
3𝑥+5
𝑥−3
≤0
The function values can
change sign at an x-intercept
of a vertical asymptote. So
3𝑥 + 1 = 𝑥 2 − 6𝑥 + 9
the regions
0 = 𝑥 2 − 9𝑥 + 8
we need to check are these:
0 = (𝑥 − 1)(𝑥 − 8)
𝑥<
𝑥 = 8 𝑜𝑟 𝑥 = 1
but 𝑥 = 8 is the only
solution that works.
−5
3
,
−5
3
< 𝑥 < 3, and 𝑥 > 3.
Using test points 𝑥 = −2, 𝑥 = 0 and 𝑥 = 4,
I find 𝑓(−2) > 0, 𝑓(0) < 0 and 𝑓(4) > 0
𝟓
(− , 𝟑)
𝟑
6. If y varies jointly as a and b and inversely as the square root of c, and y = 12 when
a = 3, b = 2, and c = 64, find y when a = 5, b = 2, and c = 25. (10 points)
Translate to a variation equation: 𝒚 = 𝒌
𝒂𝒃
𝒄
.
Use the given information to find 𝑘: 𝟏𝟐 = 𝒌
(𝟑)(𝟐)
(𝟔𝟒)
, 𝒌 = 𝟔𝟒(𝟏𝟐) ÷ 𝟔 𝒌 = 𝟑𝟐
Use 𝑘 to find y for the given values of the other variables:
𝒚 = 𝟑𝟐
(𝟓)(𝟐) 𝟔𝟒
=
= 𝟏𝟐. 𝟖
(𝟐𝟓)
𝟓