4.7 write and apply exponential & power functions p. 509 how do you write an exponential...
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4.7 Write and Apply Exponential & Power Functions
p. 509
How do you write an exponential function given two points?
How do you write a power function given two points?
Which function uses logs to solve it?
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
• 24=6b2
• 4=b2
• 2=b
a= 6/b = 6/2 = 3
So the function isY=3·2x
Write an exponential function y =ab whose graph passes through (1, 12) and (3, 108).
x
SOLUTION
STEP 1 Substitute the coordinates of the two given points into y = ab .x
12 = ab1
108 = ab3
Substitute 12 for y and 1 for x.
Substitute 108 for y and 3 for x.
STEP 2
108 = b312 b
Substitute for a in second equation.
12 b
Simplify.
Divide each side by 12.
Take the positive square root because b > 0.STEP 3
Determine that a =12 b = 12
3 = 4 so, y = 4 3 .x
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
•512=b3
•b=8 a=1/2
y=1/2 · 8x
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
Modeling with POWER functions
•y = axb
• 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 ≈ ba = 3.45y = 3.45x.535
Write a power function y = ax whose graph passes through (3, 2) and (6, 9) .
b
b2 = a 3
9 = a 6bSubstitute 2 for y and 3 for x.
Substitute 9 for y and 6 for x.
STEP 1
SOLUTION
Substitute the coordinates of the two given points into y = ax .
b
STEP 2
Solve for a in the first equation to obtain a = , and substitute this expression for a in the second equation.
23b
9 = 623b
b
9 = 2 2 b4.5 = 2
bLog 4.5 = b2
2.17 b
Substitute for a in second equation.
23b
Simplify.Divide each side by 2.Take log of each side.2
Change-of-base formula
Use a calculator.
Log 4.5 Log2 = b
STEP 3Determine that a = 0.184. So, y = 0.184x .2.173
2 2.17
Write a power function y = ax whose graph passes through the given points.
b
6. (3, 4), (6, 15)
Substitute the coordinates of the two given points into y = ax .
b
b4 = a 3
15 = a 6bSubstitute 4 for y and 3 for x.
Substitute 15 for y and 6 for x.
STEP 1
SOLUTION
STEP 2
Solve for a in the first equation to obtain a = , and substitute this expression for a in the second equation.
43b
43b15 = 6b
15 = 4 2 b
Log 3.7 = b2
Substitute for a in second equation.
43b
Simplify.
Divide each side by 4.
Take log of each side.2
= 2b15
4
3.7 = 2
1.90 b
Change-of-base formula
Use a calculator.
Log 3.7 Log2 = b
0.56820.3010= 1.9 Simplify.
STEP 3
Determine that a = 0.492. So, y = 0.492x .1.934 1.91
Biology page 284
• Find a power model for the original data.
The table at the right shows the typical wingspans x (in feet) and the typical weights y (in pounds) for several types of birds.
Biology
• 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)?
• 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
• 3-7, 15-19