Question

On a coordinate plane, an exponential function approaches y = 0 in quadrant 1 and increases into quadrant 2. It goes through (2, one-ninth), (1, one-third), (0, 1), and (negative 1, 3).
Analyze the graph of the exponential decay function.

The initial value is
.



The base, or rate of change, is
.



The domain is
.

Answer 1

Answer:

Initial value (a) = 1

Base (b) = ¹/₃

Domain = (-∞, ∞)

Step-by-step explanation:

General form of an exponential function:

  $y=ab^x$

where:

• a is the initial value (y-intercept)
• b is the base (growth/decay factor) in decimal form
• x is the independent variable
• y is the dependent variable

If b > 1 then it is an increasing function.

If 0 < b < 1 then it is a decreasing function.    

Initial value (a)

The initial value is the y-intercept.

The y-intercept is where the curve crosses the y-axis (when x = 0).  

We have been given the point (0, 1), therefore the y-intercept is 1, and so:

• Initial value (a) = 1

Base (b)

To find the value of the base (b), substitute the found value of a and one of the given points (2, ¹/₉) into the equation:

$\implies y=ab^x$

$\implies \dfrac{1}{9}=(1)b^2$

$\implies b^2=\dfrac{1}{9}$

$\implies \sqrt{b^2}=\sqrt{\dfrac{1}{9}}$

$\implies b=\dfrac{1}{3}$

Therefore:

• base = ¹/₃

Finally, the equation of the given exponential function is:

$y=\left(\dfrac{1}{3}\right)^x$

Domain

The domain of a function is the set of all possible input values (x-values).

The domain of the found exponential function is unrestricted.

Therefore:

• Domain = (-∞, ∞)

Learn more about exponential functions here:

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