5.1 Review

Which of the following equations represent exponential decay? (SELECT ALL THAT APPLY).

Answer explanation

Look at the base (this is the value raised to the exponent) in order to determine whether it is growth or decay. If the base is less than one, it is decay. If the base is greater than one, it is growth.

Which of the following represents an exponential function f(x), with a starting value of 3,240 and a 5.2% rate of increase?

•  $f\left(x\right)=3240\left(1+5.2\right)^x$  

•  $f\left(x\right)=3240\left(1-5.2\right)^x$  

•  $f\left(x\right)=3240\left(1-.052\right)^x$  

•  $f\left(x\right)=3240\left(1+.052\right)^x$  

Answer explanation

3240 represents a in y=a(b)^x, since it is the starting value. 

Since it is a rate of increase, the b should be represented by 1+r in the equation.

Convert the rate of growth percentage to a decimal: 5.2% is .052 when substituting for r. 

What are the transformations performed on the parent function $f\left(x\right)=3^x$ to get to the function  $g\left(x\right)=-\left(3\right)^{\left(x-2\right)}+4$  ? 

SELECT ALL THAT APPLY.

• reflection over the x axis

• right 2 units

• down 2 units

• right 4 units 

• up 4 units

Answer explanation

the negative a value represents a reflection over the x axis

addition or subtraction in the exponent represents left/right translations (x-2) means right 2 *it is opposite the sign

addition or subtraction outside the exponent represents up/down translations +4 outside the exponent means up 4 *it is same as the sign

What is the RANGE of the function shown in the graph?

• all real numbers

•  $\left(2,\infty\right)$  

•  $\left(0,\infty\right)$  

Which is the correct end behavior for the function graphed?

• $x\rightarrow\infty,\ y\rightarrow-\infty\ \ \ x\rightarrow-\infty,\ y\rightarrow\infty$

• $x\rightarrow\infty,\ y\rightarrow0\ \ \ \ x\rightarrow-\infty,\ y\rightarrow\infty$

• $x\rightarrow\infty,\ y\rightarrow\infty\ \ \ \ x\rightarrow-\infty,\ y\rightarrow0$

• $x\rightarrow\infty,\ y\rightarrow\infty\ \ \ \ x\rightarrow-\infty,\ y\rightarrow-\infty$

Answer explanation

As x gets large (going out to the right), the y values flatten out toward the asymptote at y=0

As x gets small (going out to the left), the y values increase (to positive infinity)

Which of the following equations represent an exponential function with an asymptote of y=2?

• $f\left(x\right)=5^{\left(x-2\right)}$

• $f\left(x\right)=5^x+2$

• $f\left(x\right)=2\left(5\right)^x$

Answer explanation

Vertical translations (up/down) will shift the parent asymptote of y=0 up or down.

Adding 2 outside of the exponent will shift the parent function & asymptote up 2 units, so the asymptote will be y=2.

A certain population of frogs is decreasing at a rate of 5.6% per year. If the initial population is 75 frogs, how many years will it take for the population to reach 50 frogs?

*graph the function & use your graphing calculator to find the intersection point

• 3 years

• 5 years

• 7 years

• 12 years

Answer explanation

enter y1= 75(1+.056)^x

enter y2=50

press 2nd, then TRACE, then select option 5- intersect

press enter with the cursor on the first curve, press enter with the cursor on the horizontal line, then press enter a 3rd time