Linear Equations Logarithms and Graphical Functions Math Quiz Problems

Please remember to show all work following standard mathematical practice:

1) Each step should show the COMPLETE expression or equation, not just a piece of it.

2) Each new step should follow logically from the step above it, following rules of algebra.

3) Each new step should be beneath the previous step.

4) The equal sign ( = ) should only connect equal numbers or expressions

•
Show work/explanation where indicated. Answers without any work may earn little, if any, credit. You may type or write
your work in your copy of the quiz, or if you prefer, create a document containing your work. Scanned work is acceptable also.
1. (4 pts) Which of these graphs represent a one-to-one function? Answer(s): ____________
(no explanation required.) (There may be more than one graph that qualifies.)
2. (6 pts) Based on data about the growth of a variety of ornamental cherry trees, the following logarithmic
model about these trees was determined:
h(t) = 5.108 ln(t) + 5.099, where t = age of tree in years and h (t) = height of tree, in feet.
(Note that “ln” refers to the natural log function) (explanation optional)
Using the model,
(a) At age 4 years, how tall is this type of ornamental cherry tree, to the nearest tenth of a foot?
(b) At age 10 years, how tall is this type of ornamental cherry tree, to the nearest tenth of a foot?
3. (4 pts) Convert to a logarithmic equation: 7x = 343.
A.
log 7 𝑥 = 343
B.
log 𝑥 7= 34
C.
3 log 7 343 = 𝑥
D.
log 𝑥 343 = 7
(no explanation required)
2. ______
4. (8 pts) Solve the equation. Check all proposed solutions. Show work in solving and in checking, and state your
final conclusion.
1 − 𝑥 = √𝑥 − 1
5. (8 pts) Do as stated:
(a)
log 3 1 =_______ (fill in the blank)
(b) Let 𝑥 = log 3
(c)
1
234
State the exponential form of the equation.
Determine the numerical value of log 3
1
234
, in simplest form. Work optional.
6. (10 pts) Let f (x) = 2×2 – 9 and g(x) = 3x + 1
(a) Find the composite function (𝑓 𝑜 𝑔)(𝑥) and simplify the results. Show work.
(b) Find (𝑓 𝑜 𝑔)(−1) . Show work.
𝑓(𝑥 ) =
7. (16 pts) Let
(a) Find f
−1
5𝑥 − 9
2𝑥 − 7
, the inverse function of f. Show work.
(b) What is the domain of f ? What is the domain of the inverse function?
(c) What is f (1) ?
(d) What is f
−1
f (1) = ______
work/explanation optional
( ____ ), where the number in the blank is your answer from part (c)?
work/explanation optional
8. (18 pts) Let f (x) = e
x–1
+ 4.
Answers can be stated without additional work/explanation.
(a) Which describes how the graph of f can be obtained from the graph of y = ex ? Choice: ________
A. Reflect the graph of y = ex across the x-axis and shift up by 4 units
B. Reflect the graph of y = ex across the y-axis and shift up by4 units.
C. Shift the graph of y = ex to the right by 1 unit and up by 4 units.
D. Shift the graph of y = ex to the left by 1 unit and up by 4 units.
(b) What is the range of f ?
(c) What is the domain of f ?
(d) What is the horizontal asymptote?
(e) What is the y-intercept? State the approximation to 2 decimal places (i.e., the nearest hundredth).
(f) Which is the graph of f ?
GRAPH A
GRAPH B
GRAPH C
GRAPH D
NONLINEAR MODELS – For the latter part of the quiz, we will explore some nonlinear models.
9. (16 pts) QUADRATIC REGRESSION
Data: On a particular summer day, the outdoor temperature was recorded at 8 times of the day, and the following table
was compiled. A scatterplot was produced, and the parabola of best fit was determined.
90
Temperature
on a Summer Day
80
Temperature (degrees)
t=
Time y = Outdoor
of day Temperature
(hour) (degrees F.)
7
52
9
67
11
73
13
76
14
78
17
79
20
76
23
61
70
60
50
y = -0.3476t2 + 10.948t – 6.0778
R² = 0.9699
40
30
20
10
0
0
4
8
12
16
20
24
Time of Day (hour)
Quadratic Polynomial of Best Fit:
y = −0.3476t2 + 10.948t − 6.0778 where t = Time of day (hour) and y = Temperature (in degrees)
REMARKS: The times are the hours since midnight. For instance, 7 means 7 am, and 13 means 1 pm.
(a) Using algebraic techniques we have learned, find the maximum temperature predicted by the quadratic model and
find the time when it occurred. Report the time to the nearest quarter hour (i.e., __:00 or __:15 or __:30 or __:45). (For
instance, a time of 18.25 hours is reported as 6:15 pm.) Report the maximum temperature to the nearest tenth of a degree.
Show algebraic work.
(b) Use the quadratic polynomial to estimate the outdoor temperature at 8:30 am, to the nearest tenth of a
degree. (work optional)
(c) Use the quadratic polynomial y = −0.3476t2 + 10.948t − 6.0778 together with algebra to estimate the time(s) of
day when the outdoor temperature y was 65 degrees.
That is, solve the quadratic equation 72 = −0.3476t2 + 10.948t − 6.0778 .
Show algebraic work in solving. State your results clearly; report the time(s) to the nearest quarter hour.