Transcript 8.7 Modeling with Exponential & Power Functions

4.7 Write and Apply Exponential &
Power Functions
p. 509
How do you write an exponential function
given two points?
Just like 2 points determine a
line, 2 points determine an
exponential curve.
An Exponential Function is in
the form of y=abx
Write an Exponential function,
y=abx whose graph goes thru
(1,6) & (3,24)
• Substitute the coordinates into y=abx to
get 2 equations.
• 1. 6=ab1
• 2. 24=ab3
• Then solve the system:
Write an Exponential function,
y=abx whose graph goes thru
(1,6) & (3,24) (continued)
• 1. 6=ab1 → a=6/b
• 2. 24=(6/b) b3
a= 6/b = 6/2 = 3
•
•
•
24=6b2
4=b2
2=b
So the function is
Y=3·2x
x
Write an exponential function y =ab whose graph
passes through (1, 12) and (3, 108).
SOLUTION
STEP 1 Substitute the coordinates of the two
given points into y = ab x .
Substitute 12 for y and 1 for x.
12 = ab 1
108 = ab 3 Substitute 108 for y and 3 for x.
STEP 2
3
108 = 12
b
b
Substitute 12
b for a in
second equation.
Simplify.
Divide each side by 12.
Take the positive square
root because b > 0.
STEP 3
12 = 4 so, y = 4 3 x.
Determine that a = 12
=
b
3
Write an Exponential function,
y=abx whose graph goes thru
(-1,.0625) & (2,32)
• .0625=ab-1
• 32=ab2
•(.0625)=a/b
•b(.0625)=a
•32=[b(.0625)]b2
•32=.0625b3
y=1/2 · 8x
•512=b3
•b=8 a=1/2
4.7 Assignment
Page 285, 3-8 all
Power Function
A Power Function is in the form of y = axb
Because there are only two constants (a and b),
only two points are needed to determine a
power curve through the points
4.7 Write and Apply Exponential &
Power Functions- Day 2
p. 509
How do you write a power function given two
points?
Which function uses logs to solve it?
Modeling with POWER functions
•y =
b
ax
• Only 2 points are
needed
• (2,5) & (6,9)
• 5 = a 2b
• 9 = a 6b
a = 5/2b
9 = (5/2b)6b
9 = 5·3b
1.8 = 3b
log31.8 = log33b
.535 ≈ b
a = 3.45
y = 3.45x.535
Write a power function y = axb whose graph
passes through (3, 2) and (6, 9) .
SOLUTION
STEP 1
Substitute the coordinates of the two given
b
points into y = ax .
2 = a 3b
Substitute 2 for y and 3 for x.
b
9=a 6
Substitute 9 for y and 6 for x.
STEP 2
2
Solve for a in the first equation to obtain a = 3b ,
and substitute this expression for a in the second
equation.
2
2 b
Substitute 3b for a in
9 = 3b 6
second equation.
9=2 2b
Simplify.
4.5 = b
Divide each side by 2.
Log2 2 4.5 = b
Take log 2 of each side.
Log 4.5 = b
Change-of-base formula
Log2
Use a calculator.
2.17 b
STEP 3
2
Determine that a = 32.17
0.184. So, y = 0.184x 2.17 .
b
Write a power function y =ax whose graph passes
through the given points.
6. (3, 4), (6, 15)
SOLUTION
STEP 1
Substitute the coordinates of the two given
b
points into y = ax .
4 = a 3b
Substitute 4 for y and 3 for x.
15 = a 6 b
Substitute 15 for y and 6 for x.
STEP 2
4
Solve for a in the first equation to obtain a = 3b, and
substitute this expression for a in the second
equation.
4
4 b
Substitute
for a in
15 = 3b 6
3b
second equation.
b
15 = 4 2
Simplify.
15 = 2 b
Divide each side by 4.
4
3.7 = 2
Log 2 3.7 = b
Take log 2 of each side.
Log 3.7 = b
Log2
0.5682
0.3010 = 1.9
1.90 b
Change-of-base formula
Simplify.
Use a calculator.
STEP 3
4
Determine that a = 31.9
0.492. So, y = 0.492x
1.91
.
Biology page 284
Biology The table at the right shows the typical
wingspans x (in feet) and the typical weights y (in
pounds) for several types of birds.
• Draw a scatter plot of the data pairs (ln x, ln y). Is a
power model a good fit for the original data pairs
(x, y)?
• Find a power model for the original data.
• How do you write an exponential
function given two points?
y = abx
• How do you write a power function
given two points?
y = axb
• Which function uses logs to solve it?
Power function y = axb
Homework
• Page 285
15-20