Let's solve for the exponential functions step by step.
### Exponential Function for Points (0, 0.5) and (1, 3)
An exponential function can be written as:
[tex]\[ y = a \cdot b^x \][/tex]
1. Substituting the first point (0, 0.5) into the equation:
[tex]\[ 0.5 = a \cdot b^0 \][/tex]
Since [tex]\( b^0 = 1 \)[/tex],
[tex]\[ 0.5 = a \][/tex]
Hence, [tex]\( a = 0.5 \)[/tex].
2. Substituting the second point (1, 3) into the equation:
[tex]\[ 3 = 0.5 \cdot b^1 \][/tex]
Simplifying,
[tex]\[ 3 = 0.5b \][/tex]
Dividing both sides by 0.5,
[tex]\[ b = 6 \][/tex]
Therefore, the exponential function that includes the points (0, 0.5) and (1, 3) is:
[tex]\[ y = 0.5 \cdot 6^x \][/tex]
### Exponential Function for Points (-1, 5) and (0.5, 40)
Similarly, we can use the form:
[tex]\[ y = a \cdot b^x \][/tex]
1. Substituting the first point (-1, 5) into the equation:
[tex]\[ 5 = a \cdot b^{-1} \][/tex]
This implies,
[tex]\[ 5 = \frac{a}{b} \][/tex]
[tex]\[ a = 5b \][/tex]
2. Substituting the second point (0.5, 40) into the equation:
[tex]\[ 40 = 5b \cdot b^{0.5} \][/tex]
Simplifying,
[tex]\[ 40 = 5b^{1.5} \][/tex]
Dividing both sides by 5,
[tex]\[ 8 = b^{1.5} \][/tex]
Taking both sides to the power of [tex]\( \frac{2}{3} \)[/tex],
[tex]\[ b = 8^{\frac{2}{3}} \][/tex]
Since [tex]\( 8 = 2^3 \)[/tex],
[tex]\[ b = (2^3)^{\frac{2}{3}} = 2^2 = 4 \][/tex]
3. Solving for [tex]\( a \)[/tex]:
[tex]\[ a = 5b = 5 \cdot 4 = 20 \][/tex]
Therefore, the exponential function that includes the points (-1, 5) and (0.5, 40) is:
[tex]\[ y = 20 \cdot 4^x \][/tex]
### Summary
1. The exponential function for the points (0, 0.5) and (1, 3) is:
[tex]\[ y = 0.5 \cdot 6^x \][/tex]
2. The exponential function for the points (-1, 5) and (0.5, 40) is:
[tex]\[ y = 20 \cdot 4^x \][/tex]
