Question

which exponential equation contains the points (0,16) and (1,6.4)? (1 point)
○ $y = 6.4^x$
○ $y = -16 \cdot 0.4^x$
○ $y = 16 \cdot 0.4^x$
○ $y = 16 \cdot 6.4^x$
graphing calculator

Explanation:

Step1: Recall exponential form

The general form of an exponential function is \( y = a \cdot b^x \), where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the base.

Step2: Use the point \((0, 16)\)

When \( x = 0 \), \( y = 16 \). Substitute into \( y = a \cdot b^x \):
\( 16 = a \cdot b^0 \). Since \( b^0 = 1 \) for any \( b
eq 0 \), we get \( a = 16 \). So the function should have \( a = 16 \), eliminating options with \( a
eq 16 \) (like \( y = 6.4^x \) and \( y = -16 \cdot 0.4^x \)).

Step3: Use the point \((1, 6.4)\)

Now we have \( y = 16 \cdot b^x \). Substitute \( x = 1 \), \( y = 6.4 \):
\( 6.4 = 16 \cdot b^1 \)
\( b = \frac{6.4}{16} = 0.4 \)

Step4: Determine the function

So the function is \( y = 16 \cdot 0.4^x \), which matches one of the options.

Answer:

\( y = 16 \cdot 0.4^x \) (the third option: \( y = 16 \cdot 0.4^x \))