Question

guided practice
write an exponential function that contains the points (1, 18) and (2, 9).
a. $y = \frac{9}{4}(2)^x$
b. $y = 36(0.5)^x$
c. $y = 9(2)^x$
d. $y = 0.5(36)^x$

Explanation:

Step 1: Recall the exponential function form

The general form of an exponential function is \( y = ab^x \), where \( a \) is the initial value and \( b \) is the base.

Step 2: Substitute the first point \((1, 18)\)

Substitute \( x = 1 \) and \( y = 18 \) into \( y = ab^x \):
\( 18 = ab^1 \), so \( ab = 18 \). Let this be Equation (1).

Step 3: Substitute the second point \((2, 9)\)

Substitute \( x = 2 \) and \( y = 9 \) into \( y = ab^x \):
\( 9 = ab^2 \). Let this be Equation (2).

Step 4: Solve for \( b \)

Divide Equation (2) by Equation (1):
\( \frac{ab^2}{ab} = \frac{9}{18} \)
Simplify: \( b = \frac{1}{2}= 0.5 \)

Step 5: Solve for \( a \)

Substitute \( b = 0.5 \) into Equation (1):
\( a(0.5)=18 \)
Solve for \( a \): \( a=\frac{18}{0.5}=36 \)

Step 6: Write the exponential function

Substitute \( a = 36 \) and \( b = 0.5 \) into \( y = ab^x \):
\( y = 36(0.5)^x \)

We can also check the second point \((2, 9)\) in this function:
When \( x = 2 \), \( y = 36(0.5)^2 = 36\times0.25 = 9 \), which matches the given point. So the correct function is \( y = 36(0.5)^x \).

Answer:

B. \( y = 36(0.5)^x \)