College Algebra

Assignment5HWM.docx

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1. (6 pts) Based on data about the growth of a variety of ornamental trees, the following logarithmic model about these trees was determined:

h(t) = 6.108 ln(t) + 6.099, 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 5 years, how tall is this type of ornamental cherry tree, to the nearest tenth of a foot?

(b) At age 15 years, how tall is this type of ornamental cherry tree, to the nearest tenth of a foot?

2. (3 pts)

Convert to a logarithmic equation: (no explanation required) 4. ___

3. (8 pts)

Solve the equation. Check all proposed solutions. Show work in solving and in checking, and state your final conclusion..

4. (8 pts)

(a) ______ (fill in the blank)

(b) Let State the exponential form of the equation.

(b) Determine the numerical value of , in simplest form. Work optional.

5. (10 pts)

Let f (x) = 2x2 – 9 and g(x) = 3+2x

(a) Find the composite function and simplify the results. Show work.

(b) Find . Show work.

6. (15 pts)

Let

(a) Find the inverse function of f. Show work.

(b) What is the domain of f (x)? What is the domain of the inverse function?

(d) What is f 1 (___ ), where the number in the blank is your answer from part (c)? work/explanation optional

7. (15 pts)

Let f (x) = e – x + 3.

Answers can be stated without additional work/explanation.

(a) Which describes how the graph of f can be obtained from the graph of y = e x ? Choice: ________

A. Shift the graph of y = e x to the right by 1 unit and shift upward by 3 units.

B. Shift the graph of y = e x to the left by 1 unit and shift upward by 3 units.

C. Reflect the graph of y = e x across the x-axis and shift upward by 3 units.

D. Reflect the graph of y = e x across the y-axis and shift upward by 3 units.

(b) What is the y-intercept?

(d) What is the range of f ?

(e) What is the horizontal asymptote?

(f) Which is the graph of f ?

GRAPH A GRAPH B GRAPH C

exp-x+3 exp-x+2 expx+2

8. (10 pts)

Use composition of functions to show that the functions and

are inverse functions.

That is, carefully show that and show that .

9. (10 pts)

Rationalize the denominator:

10. (15 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.

t = Time of day (hour)	y = Outdoor Temperature (degrees F.)
7	52
9	67
11	73
13	76
14	78
17	79
20	76
23	61

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 7:30 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 72 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.

Temperature

on a Summer Day

y = -0.3476t2 + 10.948t - 6.0778 R² = 0.9699

7 9 11 13 14 17 20 23 52 67 73 76 78 79 76 61

Time of Day (hour)

Temperature (degrees)

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212

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