Questions: Find an equation for the graph sketched below f(x)=

Solution Steps

The graph represents an exponential function of the form $f(x) = ab^x + c$, where $a$, $b$, and $c$ are constants.

The graph has a horizontal asymptote at $y = 3$. This tells us that $c = 3$. So, our equation becomes $f(x) = ab^x + 3$.

The graph intersects the y-axis at the point $(0, 1)$. Substituting these values into our equation gives us $1 = ab^0 + 3$. Since $b^0 = 1$, this simplifies to $1 = a + 3$, which means $a = -2$. Our equation is now $f(x) = -2b^x + 3$.

The graph passes through the point $(-1, 2)$. Substituting these values into the equation gives $2 = -2b^{-1} + 3$. Simplifying this, we have $-1 = -2b^{-1}$, which means $1 = 2b^{-1}$, so $b = 2$.

Final Answer: The equation of the graph is $f(x) = -2(2)^x + 3$.