Answer :
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]
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]
