Determine if the function is linear.

*x*

25

30

35

40

45

*y*

23.35

25.4

27.45

29.5

31.55

The function is linear.The function is not linear.

If the function is linear, find the equation of the line that can be used to predict the next entry to extend the table of values. (If the function is not linear, enter DNE.)

*y* =

2.05x+13.1

To determine if the function is linear, we need to check if the rate of change between each pair of \((x, y)\) values is constant.

### Step 1: Calculate the Rate of Change (Slope) Between Each Pair of Points
The slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

#### Calculate the slope between each pair of consecutive points:

1. Between \( (25, 23.35) \) and \( (30, 25.4) \):
   \[
   \frac{25.4 - 23.35}{30 - 25} = \frac{2.05}{5} = 0.41
   \]

2. Between \( (30, 25.4) \) and \( (35, 27.45) \):
   \[
   \frac{27.45 - 25.4}{35 - 30} = \frac{2.05}{5} = 0.41
   \]

3. Between \( (35, 27.45) \) and \( (40, 29.5) \):
   \[
   \frac{29.5 - 27.45}{40 - 35} = \frac{2.05}{5} = 0.41
   \]

4. Between \( (40, 29.5) \) and \( (45, 31.55) \):
   \[
   \frac{31.55 - 29.5}{45 - 40} = \frac{2.05}{5} = 0.41
   \]

Since the slope is constant (0.41) between each pair of points, the function is indeed linear.

### Step 2: Find the Equation of the Line
Now that we have verified that the function is linear, we can use the slope-intercept form \( y = mx + b \) to find the equation of the line.

The slope \( m \) is \( 0.41 \).

To find the y-intercept \( b \), we can use one of the points, such as \( (25, 23.35) \):
\[
y = mx + b
\]
\[
23.35 = (0.41)(25) + b
\]
\[
23.35 = 10.25 + b
\]
\[
b = 23.35 - 10.25 = 13.1
\]

Thus, the equation of the line is:
\[
y = 0.41x + 13.1
\]

### Conclusion
Since the function is linear, the equation that can be used to predict the next entry is:
\[
y = 0.41x + 13.1
\]

#### Additional Questions
1. How would the equation change if the slope between points varied slightly?
2. Can we use this equation to make predictions for values of \( x \) much larger than 45?
3. What is the significance of the y-intercept in this context?
4. How does the linearity of the function affect its graphical representation?
5. How can we verify the accuracy of our predicted next value?

**Tip:** To quickly check for linearity in a set of values, calculate the slope between pairs and look for consistency.

Determine if the function is linear.

x: 25, 30, 35, 40, 45
y: 23.35, 25.4, 27.45, 29.5, 31.55

The function is linear. The function is not linear.

If the function is linear, find the equation of the line that can be used to predict the next entry to extend the table of values. (If the function is not linear, enter DNE.)

y = 2.05x + 13.1

Explore how to determine if a function is linear by calculating slopes between consecutive points. This problem demonstrates the use of the slope formula and the slope-intercept form to predict future values, suitable for students in Grades 8-10.

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