Question

exponential models
guided practice
write an exponential function whose graph contains the points (1, 6) and (0, 2).
a. $y = 2(1)^x$
b. $y = 3(2)^x$
c. $y = 2(3)^x$
d. $y = 2(0.66)^x$

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 (when \( x = 0 \)) and \( b \) is the base.

Step 2: Find the value of \( a \)

We know the point \( (0, 2) \). Substitute \( x = 0 \) and \( y = 2 \) into \( y = ab^x \):

$$ 2 = ab^0 $$

Since \( b^0 = 1 \) for any \( b
eq 0 \), we get \( a = 2 \). So the function becomes \( y = 2b^x \).

Step 3: Find the value of \( b \)

We know the point \( (1, 6) \). Substitute \( x = 1 \), \( y = 6 \), and \( a = 2 \) into \( y = 2b^x \):

$$ 6 = 2b^1 $$

Simplify: \( 6 = 2b \). Divide both sides by 2: \( b = 3 \).

Step 4: Write the final function

Substitute \( a = 2 \) and \( b = 3 \) into \( y = ab^x \), we get \( y = 2(3)^x \), which corresponds to option C.

Answer:

C. \( y = 2(3)^x \)