Question

44 algebra, functions, and data analysis
in exercises 9 - 12:
a. plot the given data
b. determine if the data represents exponential or linear growth
c. determine an appropriate regression equation
d. predict the value of y when x = 15
9.

x	1	2	3	4	5	6	7	8	9	10
y	3	4	5	8	11	15	21	30	41	58

10.

x	1	2	3	4	5	6	7	8	9	10
y	8	12	15	18	20	22	26	29	33	36

11.

x	1	2	3	4	5	6	7	8	9	10
y	24	19	15	12	10	8	6	5	4	3

Explanation:

For Exercise 9:

Step1: Analyze differences/ratios

First differences: $4-3=1, 5-4=1, 8-5=3, 11-8=3, 15-11=4, 21-15=6, 30-21=9, 41-30=11, 58-41=17$ (not constant)
Ratios of consecutive y-values: $\frac{4}{3}\approx1.33, \frac{5}{4}=1.25, \frac{8}{5}=1.6, \frac{11}{8}=1.375, \frac{15}{11}\approx1.36, \frac{21}{15}=1.4, \frac{30}{21}\approx1.43, \frac{41}{30}\approx1.37, \frac{58}{41}\approx1.41$ (approaching a constant, so exponential)

Step2: Find exponential regression

Using a calculator/stat tool for $y=ab^x$, we get $a\approx2.43$, $b\approx1.35$
Equation: $y\approx2.43(1.35)^x$

Step3: Predict y at x=15

Substitute $x=15$:
$y\approx2.43(1.35)^{15}$
$(1.35)^{15}\approx51.88$
$y\approx2.43\times51.88\approx126.07$

Step1: Analyze differences/ratios

First differences: $12-8=4, 15-12=3, 18-15=3, 20-18=2, 22-20=2, 26-22=4, 29-26=3, 33-29=4, 36-33=3$ (nearly constant, linear)

Step2: Find linear regression

Using $y=mx+b$, calculate $m=\frac{n\sum xy-\sum x\sum y}{n\sum x^2-(\sum x)^2}$, $b=\frac{\sum y - m\sum x}{n}$
$\sum x=55$, $\sum y=219$, $\sum xy=1363$, $\sum x^2=385$, $n=10$
$m=\frac{10\times1363 - 55\times219}{10\times385 - 55^2}=\frac{13630-12045}{3850-3025}=\frac{1585}{825}\approx1.92$
$b=\frac{219 - 1.92\times55}{10}=\frac{219-105.6}{10}\approx11.34$
Equation: $y\approx1.92x+11.34$

Step3: Predict y at x=15

Substitute $x=15$:
$y\approx1.92\times15 + 11.34=28.8+11.34=40.14$

Step1: Analyze differences/ratios

First differences: $19-24=-5, 15-19=-4, 12-15=-3, 10-12=-2, 8-10=-2, 6-8=-2, 5-6=-1, 4-5=-1, 3-4=-1$ (not constant)
Ratios of consecutive y-values: $\frac{19}{24}\approx0.79, \frac{15}{19}\approx0.79, \frac{12}{15}=0.8, \frac{10}{12}\approx0.83, \frac{8}{10}=0.8, \frac{6}{8}=0.75, \frac{5}{6}\approx0.83, \frac{4}{5}=0.8, \frac{3}{4}=0.75$ (approaching a constant, exponential decay)

Step2: Find exponential regression

Using $y=ab^x$, we get $a\approx26.6$, $b\approx0.85$
Equation: $y\approx26.6(0.85)^x$

Step3: Predict y at x=15

Substitute $x=15$:
$y\approx26.6(0.85)^{15}$
$(0.85)^{15}\approx0.087$
$y\approx26.6\times0.087\approx2.31$

Answer:

a. Plot points (1,3), (2,4), (3,5), (4,8), (5,11), (6,15), (7,21), (8,30), (9,41), (10,58) on the grid
b. Exponential growth
c. $y\approx2.43(1.35)^x$
d. $\approx126$

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For Exercise 10: