Look at the following piecewise function. Use the expression evaluator to write the correct equation.

Steps to Solve

1. Identify the left function The left part of the graph stretches from $x = -8$ to $x = 2$. It's a line with a downward slope. We can find the equation of this line using two points: at $(-8, 4)$ and $(2, 0)$.

   • Calculate the slope ( m ): $$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 4}{2 - (-8)} = \frac{-4}{10} = -\frac{2}{5} $$

   • Use the point-slope form ( y - y_1 = m(x - x_1) ). Using point $(2, 0)$: $$ y - 0 = -\frac{2}{5}(x - 2) $$ Simplifying gives: $$ y = -\frac{2}{5}x + \frac{4}{5} $$

2. State the domain for the left function The left function is defined for the interval $x \leq 2$.

3. Identify the right function The right part of the graph stretches from $x = 2$ to $x = 6$. It's another line with a different slope. The two points we can use are $(2, 0)$ and $(6, -4)$.

   • Calculate the slope ( m ): $$ m = \frac{-4 - 0}{6 - 2} = \frac{-4}{4} = -1 $$

   • Using point-slope form with point $(2, 0)$: $$ y - 0 = -1(x - 2) $$ Simplifying gives: $$ y = -x + 2 $$

4. State the domain for the right function The right function is defined for the interval $x > 2$.