Question

Write an exponential equation that contains the points (-1, 5) and (2, 5000)

Write an exponential equation that contains the points (-2, 0.32) and (1, 0.625)

Answers

Answered by GPT-4o mini
To write exponential equations that include the given points, we use the general form of an exponential equation:

\[ y = ab^x \]

where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the growth (if \( b > 1 \)) or decay (if \( 0 < b < 1 \)) factor.

### For the points (-1, 5) and (2, 5000):

Using the points:

1. When \( x = -1 \), \( y = 5 \):
\[ 5 = ab^{-1} \quad \Rightarrow \quad 5 = \frac{a}{b} \quad \Rightarrow \quad a = 5b \quad \text{(Equation 1)} \]

2. When \( x = 2 \), \( y = 5000 \):
\[ 5000 = ab^2 \quad \text{(Equation 2)} \]

Now, substitute Equation 1 into Equation 2:

\[
5000 = (5b)b^2
\]
\[
5000 = 5b^3
\]
\[
b^3 = \frac{5000}{5} = 1000
\]
\[
b = 10
\]

Now substitute back to find \( a \):
\[
a = 5b = 5 \times 10 = 50
\]

Thus, the exponential equation is:

\[
y = 50 \times 10^x
\]

### For the points (-2, 0.32) and (1, 0.625):

Using the points:

1. When \( x = -2 \), \( y = 0.32 \):
\[ 0.32 = ab^{-2} \quad \Rightarrow \quad 0.32 = \frac{a}{b^2} \quad \Rightarrow \quad a = 0.32b^2 \quad \text{(Equation 1)} \]

2. When \( x = 1 \), \( y = 0.625 \):
\[ 0.625 = ab^1 \quad \Rightarrow \quad a = 0.625b \quad \text{(Equation 2)} \]

Now substitute Equation 1 into Equation 2:

\[
0.32b^2 = 0.625b
\]

Assuming \( b \neq 0 \), we can divide both sides by \( b \):

\[
0.32b = 0.625
\]
\[
b = \frac{0.625}{0.32} \approx 1.953125
\]

Now substitute back to find \( a \):

Using Equation 2:
\[
a = 0.625b \approx 0.625 \times 1.953125 \approx 1.220703125
\]

Thus, the second exponential equation is:

\[
y \approx 1.2207 \times (1.9531)^x
\]

Therefore, the two exponential equations you asked for are:

1. \( y = 50 \times 10^x \)
2. \( y \approx 1.2207 \times (1.9531)^x \)
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