A+ Answers

1. (6 pts) For each graph, is the graph symmetric with respect to the x-axis? y-axis? origin?(No explanation required. Just answer Yes or No to each question.)

(a) Symmetric with respect to the x-axis? _ N o__ y-axis? _ Y es _ origin? _ N o _

(b) Symmetric with respect to the x-axis? _ N o__ y-axis? _ N o__ origin? _ Y es _

2. (6 pts) Let . (no explanation required)

(a) State the zero(s) of the function. _ –6 or 6 ___

Solution:

6 – x = 0

x = 6 or

6 – –x = 0

x = –6

(b) Which of the following is true? 2. _ A __

A. f is an even function.

B. f is an odd function.

C. f is both even and odd.

D. f is neither even nor odd.

3. (6 pts) Which of the following equations does the graph represent? Show work or explanation. 3. _B__

A. B. C. D.

Solution:

Intercepts are (0, 8) and (6, 0)

Slope: m = (y1 – y2) / (x1 – x2) = (8 – 0) / (0 – 6) = 8 / – 6 = –4/3

Y intercept: b = y – mx = 8 – (–4/3)*(0) = 8

Required equation: y = mx + b y = –4x/3 + 8

4. (12 pts) Consider the points (– 4, –1) and (8, 8).

(a) Find the slope-intercept equation of the line passing through the two given points. Show work.

Solution:

Slope: m = (y1 – y2) / (x1 – x2) = (–1 – 8) / (–4 – 8) = –9 / – 12 = 3/4

Y intercept: b = y – mx = 8 – (3/4)*(8) = 2

Required equation: y = mx + b y = 3x/4 + 2

(b) Graph the line you found in (a), either drawing it on the grid in the previous problem #3, or generating the graph electronically and attaching it.

(c) Compare your line for this problem, #4, with the line in the previous problem #3. Are the two lines parallel, perpendicular, or neither parallel nor perpendicular? (The terms parallel and perpendicular are discussed on pages 166 and 167.) No explanation required – just state the answer.

Answer: Perpendicular

5. (8 pts) The number of guests g in a water park t hours after 9 am is given by g(t) = –32t2 + 224t + 263 for 0 t 8. (t = 0 corresponds to 9 am)

Find and interpret the average rate of change of g over the interval {t | 0 t 3} = [0, 3].

Show work.

Solution:

g(0) = –32t2 + 224t + 263 = –32( 0 )2 + 224( 0) + 263 = 263 and

g(3) = –32t2 + 224t + 263 = –32( 3 )2 + 224( 3) + 263 = 647

Average rate of change = [g(3) - g(0)] / [3 – 0] = (647 – 263) / 3 = 128

For every 1 hr after 9am, the number of guests in the water park increases by 128.

6. (20 pts) Ted's car has broken down and he needs it towed. A quick search reveals that two towing services offer the following arrangements:

Acme Towing: Pay a fee of $60.00, plus $2.50 per mile towed

or BeQuick: Pay a fee of $33.00, plus $4.00 per mile towed

(a) State a linear function f (x) that represents Acme Towing's total charge for towing a vehicle x miles.

Answer:

f (x) = 2.50(x) + 60

(b) State a linear function g(x) that represents BeQuick Towing's total charge for towing a vehicle x miles.

Answer:

g (x) = 4.0(x) + 33

(c) Ted needs to have his car towed 22 miles, as cheaply as possible. Which towing service should he choose? Show work/explanation.

Solution:

If x = 22,

f (22) = 2.50(22) + 60 = $115

and

g (22) = 4.0(22) + 33 = $121

So the cheaper towing service is Acme Towing

(d) For what distance towed is the total towing charge exactly the same for both towing services? Show algebraic work/explanation.

Solution:

f (x) = g (x)

2.50(x) + 60 = 4.0(x) + 33 60 – 33 = 4x – 2.5x 27 = 1.5x x = 18 miles

(e) Fill in the blanks:

The BeQuick towing service is the cheaper option if the vehicle is towed __ less ___(choose less or more ) than __ 22 ____ (enter number) miles.

A graph of y = f (x) follows. No formula for f is given. 7. __ D __

Which graph (A, B, C, or D) represents the graph of y = f (x + 1) 2 ?

EXPLAIN YOUR CHOICE. (Grid supplied for scratch work; You are NOT required to submit your own graph)

y = f (x)

( 2 4 -4 -2 -4 2 4 - 2 ) ( 2 4 -4 -2 -4 2 4 - 2 )

GRAPH A (below) GRAPH B (below)

( 2 4 -4 -2 -4 2 4 - 2 ) ( 2 4 -4 -2 -4 2 4 - 2 )

GRAPH C (below) GRAPH D (below)

( 2 4 -4 -2 -4 2 4 - 2 ) ( 2 4 -4 -2 -4 2 4 - 2 )

Given the graph of the function f(x), then the graph of f(x + a) – b will move “a” units leftward, and “b” units downward

8. (20 pts, no explanation required)

Consider the graph of the piecewise function y = f (x) pictured below.

(a) State the value of f (2).

Answer:

f (2) = 1

(b) State the x-intercept(s), if any.

Answer:

x-intercepts are (5, 0), (0, 0) and (3, 0)

(c) State the y-intercept(s), if any.

Answer:

y-intercept is (0, 0)

(d) State the domain of the piecewise function.

Answer:

Domain: [6, 5]

(e) State the range of the piecewise function.

Answer:

Range: [1, 2]

(f) State the interval(s) on which the function is increasing. That is, for what x-values is the function increasing?

Answer:

[1, 2] and [3, 5]

(g) State the interval(s) on which the function is decreasing. That is, for what x-values is the function decreasing?

Answer:

[6, 4] and [2, 3]

9. (14 pts Life expectancy at birth is the estimated lifespan of a baby born in a particular year (given the conditions of that time period). Based on data retrieved from http://www.indexmundi.com/facts/united-states/life-expectancy-at-birth the following chart of U.S. life expectancy for females has been prepared.

The regression line is y = 0.151x – 222.26, where x = birth year and y = U.S. female life expectancy, in years. The value of r2 is 0.9366.

(a) Use the regression line to estimate the U.S. life expectancy of a female baby born in 1970, to the nearest tenth of a year. Show some work.

Solution:

If x = 1970, y = 0.151x – 222.26 y = 0.151 (1970) – 222.26 = 75.21 or approx. 75.2 yrs

(b) Use the regression line to estimate the U.S. life expectancy of a female baby born in 2016, to the nearest tenth of a year. Show some work.

Solution:

If x = 2016, y = 0.151x – 222.26 y = 0.151 (2016) – 222.26 = 82.156 or approx. 82.2 yrs

(c) What is the slope of the regression line and what are the units of measurement? In a sentence, interpret what the slope is telling us, in the context of this real-world application.

Answer:

The slope of the regression line is 0.151, and the unit is in years. The slope means that for every year, the US female life expectancy at birth increases by 0.151 years

(d) What is the value of the correlation coefficient, r ? Also, interpret its value: Looking at the graph and the size of r, do you judge the strength of the linear relationship to be very strong, moderately strong, somewhat weak, or very weak?

Answer:

The value of r2 is 0.9366 and this means that 93.66% of the variations or changes in the dependent variable (life expectancy) can be explained and attributed by the variations or changed in the independent variable (birth year). Since the value of r2 = 0.9366, then the correlation coefficient is = (0.9366) = 0.96778 and this value of correlation coefficient shows a very strong positively linear relationship between the two variables.