Understanding Relations and Functions

Math I - Lessons 3.1/3.2 Review

Exponent Rules

Exponential Growth and Decay

Exponent Graphs and Tables

Exponential Functions

Multiple Choice

Which of the following is the simplified form of the expression

$$\frac{\left(3x^2\right)^3}{9x^3}$$  ?

• $$x^3$$  

• $$x^2$$  

• $$3x^3$$  

• $$3x$$  

Multiple Choice

Simplify fully:

$$45x^4\cdot\left(3x^2\right)^{-2}$$  

• 5

• $$5x^0$$  

• $$15x^8$$  

• $$9x$$  

Exponent Rules

• Product of Powers:

   $x^a\cdot x^b=x^{\left(a+b\right)}$  

• Quotient of Powers:  $\frac{x^a}{x^b}=x^{\left(a-b\right)}$  

• Power to a Power:  $\left(x^a\right)^b=x^{\left(ab\right)}$  

• Power of a Product:  $\left(xy\right)^a=x^ay^a$  

• Zero Property:  $x^0=1\ \left(x\ \ne0\right)$  

• Negative Exponent:  $x^{-a}=\frac{1}{x^a}$  

Multiple Select

Which of the following representations show relationships of exponential growth?

• $$f\left(x\right)=3\left(4\right)^x$$

• A bacterial population, starting from a single cell, doubles in size every hour

• $$f\left(x\right)=25\left(0.85\right)^x$$

Multiple Select

Which of the following representations show a relationship of exponential decay?

• $$f\left(x\right)=12\left(1.45\right)^x$$

• A 500mg dose of acetaminophen loses 30% of its concentration strength in the bloodstream each hour

• A retirement portfolio that increases in value by 3% every year

Exponential Growth and Decay

• GROWTH = values increase over time; exponential rate b > 1

• DECAY = values decrease over time; exponential rate 0 < b < 1

Multiple Choice

Which of the following functions matches the graph shown here?

• $$f\left(x\right)=3\left(2\right)^x$$

• $$f\left(x\right)=3\left(0.2\right)^x$$

• $$f\left(x\right)=2\left(3\right)^x$$

• $$f\left(x\right)=2\left(0.3\right)^x$$

Exponent Graphs

• Need to find values for a (initial) and b (multiplier)

• To find a: look for y-intercept

• To find b: look at x = 1

• Plug in values, solve for b OR find multiplier between outputs

Multiple Choice

Which of the following functions matches with the table shown here?

• $$f\left(x\right)=0.5\left(20\right)^x$$

• $$f\left(x\right)=20\left(2\right)^x$$

• $$f\left(x\right)=20\left(0.5\right)^x$$

• $$f\left(x\right)=1.25\left(5\right)^x$$

Exponent Tables

• Finding a: Look for x = 0

• Finding b: Look for product pattern between output values (work backwards by division)

Open Ended

A $5,000 savings account earns 1.75% interest per year. If Marisa starts the savings account for her son at birth, how much will be in the account when her son turns 18 years old?

Building Exponential Functions

•  $y=ab^x$  

• a = starting value

• b = multiplier 

• When rate is a percentage, convert to a decimal and ADD to 1 (growth) or SUBTRACT from 1 (decay) for b

Open Ended

A 750mg sample of radioactive plutonium decays (losing its radioactivity) by half every day. How much radioactive plutonium will remain at the end of a week?