Exponent/ Logarithm Quiz Review

Exponent/ Logarithm Quiz Review

Crash Course

Basics of exponential functions.

• General form:  $y=a\cdot b^x$  

• The 'a': This is the y-intercept. If it is positive, your graph is above the x-axis. If it is negative, your graph will be below the x-axis.

• The 'b': This determines if the graph is going to be growing or decaying. If b>1, you have exponential growth. If 0<b<1, you have exponential decay.

Multiple Choice

Is the following function an example of decay or growth?

f(x) = 2(1.25)^x

• Exponential Growth

• Exponential Decay

Multiple Choice

Is the following function and example of decay or growth?

f(x)=2(0.85)^x

• Exponential Decay

• Exponential Growth

Multiple Choice

Identify the y-intercept (initial value) of the function f(x)=2(4)^x.

• 2
• 4
• (2)^x
• 8

Multiple Choice

Is the pictured graph growth, decay, or linear or none?  

• Growth
• Decay
• Linear
• None

Multiple Choice

Is the pictured graph growth, decay, or linear or none?  

• Growth
• Decay
• Linear
• None

Basics of exponential functions. (continued)

• Domain: For exponential functions, it will always be  $\left(-\infty,\infty\right)$  !

• Range: Look at the horizontal asymptote! In this picture it is y=0, meaning all possible y values will be  $\left(0,\infty\right)$  .

Basics of exponential functions. (continued)

• Asymptotes: Never vertical asymptotes. Always a horizontal asymptote. Where does the graph approach? (y=0)

• x-intercept: Not every exponential function will have one!

• y-intercept: when x=0, but a quick check is remember the 'a' in  $y=a\cdot b^x$  is the y intercept.

Basics of exponential functions. (continued)

• End behavior: Look at the graph as it approaches  $-\infty$   and  $\infty$   for its x values! 

•  $\lim_{x\rightarrow-\infty}\ f\left(x\right)=0\$  

•  $\lim_{x\rightarrow\infty}\ f\left(x\right)=\infty$  

Multiple Choice

What is the range of this function?

• (-∞,0)

• (0,∞)

• (-∞,4)

• (4,∞)

Multiple Choice

Identify the y-intercept (initial value) in the function f(x)=13(.27)^x

• (.27)^x
• .27
• 13
• 3.51

Multiple Choice

What is the x-intercept of the exponential function?

• (0,1)

• (0,0)

• There is not one

Transformations of Graphs

• If you add a value to the whole function (i.e.  $f\left(x\right)=2^x+1$  ) then your graph moves upward.

• If you subtract a value to the whole function (i.e.  $f\left(x\right)=2^x-1$  ) then your graph moves downward.

Transformations of Graphs

• If you add a value to the exponent (i.e.  $f\left(x\right)=2^{\left(x+1\right)}$  ) then your graph moves to the left.

• If you subtract a value to the exponent (i.e.  $f\left(x\right)=2^{\left(x-1\right)}$  ) then your graph moves to the right.

Applications

• Growth formula:   $A=P\left(1+r\right)^t$  

• Decay Formula:   $A=P\left(1-r\right)^t$  

• Compound interest (they will say explicitly 'compound' in the question with the rate at which it is compounding):   $A=\ P\left(1+\frac{r}{n}\right)^{tn}$  

• Continous compounding:  $A=Pe^{rt}$  

Multiple Choice

If $1,000 is invested at 16% interest, compounded continuously, for five years, what is the ending balance?

• $1,225,54
• $2,225.54
• $22,255.40
• $225.54

Multiple Choice

Suppose a culture of bacteria begins with 5000 cells and dies by 30% each year. Write an equation that represents this situation.

• y=5000(0.7)^x
• y=30(5000)^x
• y=5000(1.3)^x
• y=5000x^x

Multiple Choice

Most automobiles depreciate as they get older. Suppose an automobile that originally costs $14,000 depreciates by 20% of its value every year.

What is the value of this automobile after 5 years?

• $264,539.52

• $4.48

• $4,587.52

• $34,836.48

Graphing exponentials by hand.

• ALWAYS HAS A HORIZONTAL ASYMPTOTE.

• Diagnose the 'a' (above or below x-axis)

• Diagnose the 'b' (growing or decaying)

• Plug in x=0 and x=1 to find two guiding points.

Graphing logarithms by hand.

• ALWAYS HAS A VERTICAL ASYMPTOTE

• Consider the vertical asymptote and how that will shape your graph.

• Plug in two easy x-values like x=1 and x=2, and evaluate for the log to find the coordinate.

• Use these coordinates as a guide for your line.