https://file.aiquickdraw.com/mathbot/file/172401727403670gfcc4g.jpg

Let's analyze the table in the new image:

| \( x \) | 1  | 2  | 3  | 4  |
|--------|----|----|----|----|
| \( y \) | 4  | 1  | -2 | -5 |

To find the function rule, let's first examine if there is a pattern. We can observe that the values of \( y \) are decreasing in a regular pattern. Specifically:

- From \( 4 \) to \( 1 \) (decrease by 3),
- From \( 1 \) to \( -2 \) (decrease by 3),
- From \( -2 \) to \( -5 \) (decrease by 3).

This constant rate of change suggests that the function is linear, in the form \( y = mx + b \), where:
- \( m \) is the slope (rate of change), and
- \( b \) is the y-intercept (value of \( y \) when \( x = 0 \)).

### Step 1: Find the slope \( m \)
The slope \( m \) is the change in \( y \) divided by the change in \( x \):
\[
m = \frac{\Delta y}{\Delta x} = \frac{-5 - (-2)}{4 - 3} = \frac{-3}{1} = -3
\]

So, the slope is \( -3 \).

### Step 2: Use a point to find the y-intercept \( b \)
Now that we have the slope, we can use one of the points from the table to solve for \( b \). Let's use the point \( (1, 4) \).

Using the equation \( y = mx + b \), substitute \( x = 1 \), \( y = 4 \), and \( m = -3 \):

\[
4 = -3(1) + b \quad \Rightarrow \quad 4 = -3 + b \quad \Rightarrow \quad b = 7
\]

### Final Function Rule
The function rule is:
\[
y = -3x + 7
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How can you graph the function \( y = -3x + 7 \) based on the table?
2. What happens to the slope if the differences in \( y \) values aren't constant?
3. How can you determine the y-intercept if the table doesn’t provide \( x = 0 \)?
4. How does the slope \( m = -3 \) affect the steepness and direction of the graph?
5. How would the function rule change if the table had quadratic data?

**Tip**: For linear functions, always check if the differences between consecutive \( y \)-values are constant to confirm a linear relationship.

Learn how to find the function rule of a linear equation using the slope-intercept form. This problem involves analyzing a table of values to determine the slope and y-intercept. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation