Question

quiz 8 write the equation of a graph
f(x) =

a =

b =

asymptote =

Explanation:

Step1: Identify function type

This is an exponential function, so we use the form $f(x) = ab^x$, where $a$ is the y-intercept, $b$ is the growth factor, and there is a horizontal asymptote.

Step2: Find $a$ (y-intercept)

The graph crosses the y-axis at $(0,2)$. Substitute $x=0$, $f(x)=2$ into $f(x)=ab^x$:
$2 = ab^0$
Since $b^0=1$, $a=2$.

Step3: Find $b$ (growth factor)

Use the point $(1,4)$. Substitute $a=2$, $x=1$, $f(x)=4$ into $f(x)=ab^x$:
$4 = 2b^1$
Solve for $b$: $b = \frac{4}{2}=2$

Step4: Determine asymptote

Exponential functions of the form $ab^x$ have a horizontal asymptote at $y=0$, as the graph approaches but never touches $y=0$ as $x\to-\infty$.

Step5: Write full function

Substitute $a=2$ and $b=2$ into the exponential form: $f(x)=2(2)^x=2^{x+1}$

Answer:

$a=2$
$b=2$
asymptote: $y=0$
$f(x)=2(2)^x$ (or $f(x)=2^{x+1}$)