https://file.aiquickdraw.com/mathbot/file/1729277065165jz0h8jfo.jpg

Let's solve the problem step by step based on the given image.

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### (a) Rate of change for each function

#### **Rate of change for Function 1 (Table)**
The rate of change between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated by:

\[
\text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's pick two points from the table.  
For example, using points \((-1, 3)\) and \((2, -5)\):

\[
\text{Rate of Change} = \frac{-5 - 3}{2 - (-1)} = \frac{-8}{3}
\]

So, the **rate of change for Function 1** is:

\[
\frac{-8}{3} \, \text{or} \, -\frac{8}{3}
\]

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#### **Rate of change for Function 2 (Graph)**
Choose two points from the graph. From the image, it looks like the points \((0, 2)\) and \((2, -4)\) are clear.

\[
\text{Rate of Change} = \frac{-4 - 2}{2 - 0} = \frac{-6}{2} = -3
\]

So, the **rate of change for Function 2** is:

\[
-3
\]

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### (b) Which function has the greater rate of change?

Since we are comparing the absolute values of the rates of change:

- Function 1: \(\left| -\frac{8}{3} \right| \approx 2.67\)  
- Function 2: \(\left| -3 \right| = 3\)

**Function 2** has the greater rate of change.

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### Final Answer
- **(a)**  
  - Function 1: \(-\frac{8}{3}\)  
  - Function 2: \(-3\)
  
- **(b)**  
  **Function 2** has the greater rate of change.

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Do you need any further details or explanations?

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### Related Questions:
1. How do you determine the slope from a graph?
2. Why is the slope the same between any two points on a line?
3. Can the rate of change be positive? What does it represent?
4. How do different slopes affect the steepness of a line?
5. How would the rates of change change if the graph shifted vertically?

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**Tip:**  
Always compare the absolute values of rates of change to determine which function grows or declines faster.

Use the table and the graph to answer the questions. (a) What is the rate of change for each function? (b) Which function has the greater rate of change?

Learn how to determine the rate of change for two functions using a table and a graph. This problem covers key algebraic concepts related to the slope of a line and rate of change calculations. Suitable for students in Grades 8-10.

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