Look at this table:
x	y
5	
124
5
6	
149
5
7	
174
5
8	
199
5
9	
224
5
Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.

Let's analyze the given data:

| \( x \) | \( y \) |
|----|----|
| 5  | 124 |
| 6  | 149 |
| 7  | 174 |
| 8  | 199 |
| 9  | 224 |

We will determine whether the data follows a **linear** or **exponential** pattern.

### Step 1: Check for Linearity
A function is linear if it has a constant rate of change (\( m \)), meaning the difference between consecutive \( y \)-values should be the same.

Calculating the differences:

\[
149 - 124 = 25
\]

\[
174 - 149 = 25
\]

\[
199 - 174 = 25
\]

\[
224 - 199 = 25
\]

Since the differences are constant, the data follows a **linear function**.

### Step 2: Find the Linear Equation
The linear equation is of the form:

\[
y = mx + b
\]

where:
- \( m \) is the slope (rate of change), which we calculated as \( 25 \).
- \( b \) is the y-intercept, which we determine by using one of the points.

Using the point \( (5,124) \):

\[
124 = 25(5) + b
\]

\[
124 = 125 + b
\]

\[
b = -1
\]

Thus, the linear function is:

\[
y = 25x - 1
\]

### Step 3: Verify if the Function Fits
Checking for other points:

- \( x = 6 \): \( y = 25(6) - 1 = 149 \) ✅
- \( x = 7 \): \( y = 25(7) - 1 = 174 \) ✅
- \( x = 8 \): \( y = 25(8) - 1 = 199 \) ✅
- \( x = 9 \): \( y = 25(9) - 1 = 224 \) ✅

Since all values match, the equation **\( y = 25x - 1 \) correctly models the data**.

---

Would you like me to explain this in more detail?  

Here are five related questions to expand on this concept:
1. How do you determine if data follows an exponential function?
2. What does the slope represent in a linear function?
3. How can you find the equation of a line given any two points?
4. What happens when the y-intercept is negative?
5. How do you apply linear equations in real-world scenarios?

### **Tip:**  
To quickly check if data follows an **exponential pattern**, calculate the ratios between consecutive \( y \)-values instead of their differences.

Look at this table:
x	y
5	124
5
6	149
5
7	174
5
8	199
5
9	224
5
Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.

Learn how to write a linear function to model data with a constant rate of change. The problem involves calculating the slope and y-intercept from given data points, leading to the equation y = 25x - 1. Suitable for students in Grades 7-9.

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