Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.

Find a formula for the exponential function passing through the points [tex]\left(-3, \frac{5}{64}\right)[/tex] and [tex](1, 20)[/tex].

[tex]y =[/tex]


Sagot :

To find the exponential function of the form \( y = a \cdot b^x \) that passes through the points \((-3, \frac{5}{64})\) and \((1, 20)\), we need to determine the constants \(a\) and \(b\). Here's a step-by-step process:

1. Set up the system of equations:
We have two points and we need to set up two equations using the exponential function form \(y = a \cdot b^x\).

For the point \((-3, \frac{5}{64}\)):
[tex]\[ \frac{5}{64} = a \cdot b^{-3} \][/tex]

For the point \((1, 20)\):
[tex]\[ 20 = a \cdot b^1 \][/tex]

2. Solve for \(a\) using the second equation:
[tex]\[ 20 = a \cdot b \][/tex]
[tex]\[ a = \frac{20}{b} \][/tex]

3. Substitute \(a\) in the first equation:
[tex]\[ \frac{5}{64} = \left(\frac{20}{b}\right) \cdot b^{-3} \][/tex]
Simplify the equation:
[tex]\[ \frac{5}{64} = 20 \cdot b^{-4} \][/tex]
[tex]\[ \frac{5}{64} = \frac{20}{b^4} \][/tex]

4. Solve for \(b\):
[tex]\[ \frac{5}{64} \cdot b^4 = 20 \][/tex]
[tex]\[ b^4 = 20 \cdot \frac{64}{5} \][/tex]
[tex]\[ b^4 = 256 \][/tex]
[tex]\[ b = \sqrt[4]{256} \][/tex]
[tex]\[ b = 4 \][/tex]

5. Solve for \(a\) using \(b = 4\):
Substitute \(b\) back into the equation \(a = \frac{20}{b}\):
[tex]\[ a = \frac{20}{4} \][/tex]
[tex]\[ a = 5 \][/tex]

6. Write the exponential function:
Now that we have \(a = 5\) and \(b = 4\), the exponential function is:
[tex]\[ y = 5 \cdot 4^x \][/tex]

So, the exponential function that passes through the points \((-3, \frac{5}{64})\) and \((1, 20)\) is:
[tex]\[ y = 5 \cdot 4^x \][/tex]