Warm up
In 1991 the population of Humpback
whales was 5263. The Humpback
population decreases at an annual rate
of 1.5%.
What is the exponential equation
modeling this decay?
What will the Humpback population be
in 2012?
Warm up Answer
• Equation:
𝑦 = 5263 1 − .015
• Number of whales in 2012:
𝑦 = 5263 1 − .015
y= 3831 whales
21
𝑥
Rational Exponents and Roots
Remember:
• What is the difference between a Power
function and an Exponential function?
• Power has variable in the base and
exponential has variable in the exponent.
• (you need to know this for #1 on the
homework)
Definition of Rational Exponents
So….
•
𝑥=𝑥
•
5
•
3
1
2
𝑏=𝑏
1
5
𝑐4 = 𝑐
4
3
3
OR
• Also remember
1
2
∙
𝑐
4
2
1
=𝑐
4
3
= 1. So,
reciprocals of fractions cancel each other
out.
Example 1
Solution
•
3
𝑐
8
=𝑐
8
3
Rewrite with a fraction
8
3
• 𝑐 =47
• 𝑐
8
3
3
8
Set equal to original value
3
8
= 47
Raise both sides to the power
of the reciprocal to cancel out the exponent
• 𝑐 = 4.24
simplify.
3
8
Put 47 in the calculator to
Example 2: You try!
Answer:
352.33
Example 3: what if there is a
coefficient?
3
4𝑥 2 + 8.5 = 19.8
Answer
•
3
4𝑥 2 + 8.5 = 19.8
•
3
4𝑥 2 = 11.3
• 4𝑥 2
1
3
= 11.3
2
Original
Subtract 8.5
Change cube-root to a fraction
• 4𝑥 = 1442.897
Cube both sides to cancel
• 𝑥 2 = 360.72
Divide by 4
• 𝑥 = 18.99
Raise both sides to the
1
2
1
3
Remember from way back when…
• The general equation for transformation of
functions…
𝑓 𝑥 =− 𝑥−ℎ +𝑘
Example 4
3
4
• The original equation in black is 𝑦 = 𝑥 . Write
an equation for the shifted function in red.
Answer
𝑓 𝑥 = −𝑥 − 5
3
4
+1
Homework
•
•
•
•
•
5.3 Worksheet
1 all
2 a-c
3 a-f
4 a-c