To find the exponential function that passes through the points [tex]\((1, 7)\)[/tex] and [tex]\((3, 28)\)[/tex], we are looking for a function of the form [tex]\(y = a \cdot b^x\)[/tex]. Here is a step-by-step solution:
1. Define the function form: The function is [tex]\(y = a \cdot b^x\)[/tex].
2. Set up the equations using the given points:
- For the point [tex]\((1, 7)\)[/tex]:
[tex]\[
7 = a \cdot b^1
\][/tex]
- For the point [tex]\((3, 28)\)[/tex]:
[tex]\[
28 = a \cdot b^3
\][/tex]
3. Solve for [tex]\(b\)[/tex]:
- Start by isolating [tex]\(a\)[/tex] in the first equation:
[tex]\[
a = \frac{7}{b}
\][/tex]
- Substitute [tex]\(a\)[/tex] from the first equation into the second equation:
[tex]\[
28 = \left(\frac{7}{b}\right) \cdot b^3
\][/tex]
- Simplify and solve for [tex]\(b\)[/tex]:
[tex]\[
28 = 7 \cdot b^2
\][/tex]
[tex]\[
b^2 = \frac{28}{7} = 4
\][/tex]
[tex]\[
b = 2
\][/tex]
4. Solve for [tex]\(a\)[/tex]:
- Substitute [tex]\(b = 2\)[/tex] back into the first equation:
[tex]\[
7 = a \cdot 2
\][/tex]
[tex]\[
a = \frac{7}{2} = 3.5
\][/tex]
5. Form the exponential function:
- Now that we have [tex]\(a = 3.5\)[/tex] and [tex]\(b = 2\)[/tex], we can write the exponential function as:
[tex]\[
y = 3.5 \cdot 2^x
\][/tex]
Therefore, the exponential function that passes through the points [tex]\((1, 7)\)[/tex] and [tex]\((3, 28)\)[/tex] is:
[tex]\[ y = 3.5 \cdot 2^x \][/tex]
