The assignment visible in the image asks for help in graphing linear and exponential functions based on the given values.

Problem details:

• Beginning value (y-intercept): $-9$
• Change at each stage (slope): $\frac{3}{8}$

Required:

• Write the equation of the function.
• Graph the function.

Step 1: Writing the Equation

This looks like a linear function because it specifies a constant change (slope). The general form of a linear function is: $f(x) = mx + b$ Where:

• $m$ is the slope ($\frac{3}{8}$),
• $b$ is the y-intercept (starting value, $-9$).

So, the equation becomes: $f(x) = \frac{3}{8}x - 9$

Step 2: Graphing

For this function:

• The y-intercept is $-9$, so the graph passes through the point $(0, -9)$.
• The slope is $\frac{3}{8}$, meaning for every 8 units moved right, the function value increases by 3 units.

To plot the graph, you can start at $(0, -9)$ and apply the slope to find additional points.

Do you need help with more details on how to graph it or have any questions?

Here are five related questions to explore:

1. How can you tell if a function is linear or exponential based on its description?
2. What is the significance of the slope in a linear function?
3. How would the graph change if the slope was negative?
4. How do you calculate the slope given two points?
5. Can you interpret the function's graph in terms of real-world contexts?

Tip: Always check if the rate of change is constant. If it's constant, the function is linear; if it's multiplicative, it's exponential.