Question

homework assignment 6.1 exponential functions
due sunday by 11:59pm points 12 submitting an external tool
homework assignment 6.1 exponential functions
score: 5/12 answered: 5/12
question 6
find a formula for the exponential function passing through the points (-3, 4/125) and (1, 20).
y =
question help: video read written example

Explanation:

Step1: Write the general form of an exponential function

The general form of an exponential function is \(y = ab^{x}\).

Step2: Substitute the given points into the general form

For the point \((- 3,\frac{4}{125})\), we have \(\frac{4}{125}=ab^{-3}\), which can be rewritten as \(\frac{4}{125}=\frac{a}{b^{3}}\), so \(a=\frac{4}{125}b^{3}\).
For the point \((1,20)\), we have \(20 = ab^{1}\), so \(a=\frac{20}{b}\).

Step3: Equate the two expressions for \(a\)

Set \(\frac{4}{125}b^{3}=\frac{20}{b}\).
Multiply both sides by \(125b\) to get \(4b^{4}=2500\).
Then \(b^{4} = 625\), so \(b = 5\) (since \(b>0\) for an exponential function \(y = ab^{x}\)).

Step4: Find the value of \(a\)

Substitute \(b = 5\) into \(a=\frac{20}{b}\), we get \(a = 4\).

Answer:

\(y = 4\times5^{x}\)