The question asks to find a formula for each function graphed in parts a and b of the figure.

Part a:

The points given are approximately:

• (-1, 4)
• (2, 1)

These suggest a linear relationship. The formula for a linear equation is $y = mx + b$, where $m$ is the slope, and $b$ is the y-intercept.

To find the slope $m$, we use: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 4}{2 - (-1)} = \frac{-3}{3} = -1$

Now, using the point-slope form of the equation, we can plug in one of the points (for example, $(2, 1)$) to find the y-intercept $b$: $y = -1 \cdot 2 + b = 1 \implies b = 3$

Thus, the equation of the line is: $y = -x + 3$

Part b:

The points given are approximately:

• (-3, -1)
• (1, -3)
• (3, 3)

This looks like a piecewise function:

1. From $(-3, -1)$ to $(1, -3)$, calculate the slope: $m_1 = \frac{-3 - (-1)}{1 - (-3)} = \frac{-2}{4} = -\frac{1}{2}$ Using the point $(-3, -1)$ to find the equation for this segment: $y = -\frac{1}{2}(x + 3) - 1$ This simplifies to: $y = -\frac{1}{2}x - \frac{5}{2}$

2. From $(1, -3)$ to $(3, 3)$, calculate the slope: $m_2 = \frac{3 - (-3)}{3 - 1} = \frac{6}{2} = 3$ Using the point $(1, -3)$: $y = 3(x - 1) - 3$ This simplifies to: $y = 3x - 6$

Thus, the piecewise function is: