Write an exponential function that has a y-intercept of -4 and passes through the points (1, -1.6) and (2, -0.64).

Steps to Solve

1. Define the exponential function

The general form of an exponential function is given by: $$ f(x) = ab^x $$ where $a$ is the y-intercept and $b$ is the base. In this case, since the y-intercept is $-4$, we have: $$ f(x) = -4b^x $$

2. Plug in the first point (1, -1.6)

Using the point $(1, -1.6)$, plug in $x = 1$ and $f(x) = -1.6$ into the function: $$ -1.6 = -4b^1 $$ This simplifies to: $$ -1.6 = -4b $$ Now, solve for $b$: $$ b = \frac{-1.6}{-4} = 0.4 $$

3. Plug in the second point (2, -0.64)

Now use the second point $(2, -0.64)$, substituting $x = 2$ and $f(x) = -0.64$ into the function: $$ -0.64 = -4b^2 $$ We now substitute $b = 0.4$ into the equation: $$ -0.64 = -4(0.4)^2 $$ Calculate $(0.4)^2$: $$ (0.4)^2 = 0.16 $$ Now substitute this back: $$ -0.64 = -4(0.16) $$ This simplifies to: $$ -0.64 = -0.64 $$ Since this is a true statement, the value of $b$ is consistent.

4. Write the final function

Now that we have found both $a$ and $b$, we can write the final exponential function: $$ f(x) = -4(0.4)^x $$