https://file.aiquickdraw.com/mathbot/file/1726209887886ybwm5t7n.jpg

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:
\[
y = \begin{cases} 
-\frac{1}{2}x - \frac{5}{2}, & \text{for } -3 \leq x \leq 1 \\
3x - 6, & \text{for } 1 \leq x \leq 3 
\end{cases}
\]

Do you need further explanation on any part of this solution?

Here are 5 follow-up questions you might find useful:
1. How do you find the slope between two points?
2. What is the point-slope form of a line, and how is it used?
3. What is a piecewise function, and how do you identify it from a graph?
4. How do you solve for the y-intercept of a line when given a point and the slope?
5. How do you write an equation for a line segment in a piecewise function?

**Tip**: When working with piecewise functions, carefully examine the changes in slope at each segment to identify breaks in the graph!

Find a formula for each function graphed below, referring to part a and part b of the image.

Learn how to find formulas for graphed functions using linear equations and piecewise functions. The problem involves calculating slopes, determining y-intercepts, and constructing equations for segments. Suitable for high school students in Grades 9-12.

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