Question

what is the y-intercept? a=__________
what is the ratio? b=__________
what is the exponential equation? f(x)=__________

1. a=3; b=2

f(x)=3·2^x

1. a=3; b=2

f(x)=2·3^x

1. a=3; b=2 f(x)=6^x
2. a=1; b=6 f(x)=6^x

Explanation:

Step1: Identify y-intercept (A)

The y-intercept of an exponential function $f(x)=A \cdot B^x$ is the value when $x=0$. For $x=0$, $f(0)=A \cdot B^0 = A$. From the graph, when $x=0$, the function value is 3, so $A=3$.

Step2: Determine growth ratio (B)

Use a second point on the graph. When $x=1$, $f(1)=6$. Substitute $A=3$, $x=1$, $f(1)=6$ into $f(x)=A \cdot B^x$:
$6 = 3 \cdot B^1$
Solve for $B$: $B=\frac{6}{3}=2$

Step3: Write exponential equation

Substitute $A=3$ and $B=2$ into the standard form $f(x)=A \cdot B^x$:
$f(x)=3 \cdot 2^x$

Answer:

1. A=3; B=2, $f(x)=3 \cdot 2^x$