Question

the table represents some points on the graph of a linear function.
write a function that represents this relationship.
x: -4,-2,6
y: 10,7,-5
y=_____

Answer 1

To determine the function \( y = mx + b \) that represents the relationship between \( x \) and \( y \), we need to find the slope \( m \) and the y-intercept \( b \).

Given the points:
- \((-4, 10)\)
- \((-2, 7)\)
- \( (6, -5) \)

### Step 1: Calculate the Slope (m)

We can calculate the slope \( m \) using two points. Let's use the points \((-4, 10)\) and \((-2, 7)\).

The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points \((-4, 10)\) and \((-2, 7)\):

\[
m = \frac{7 - 10}{-2 - (-4)} = \frac{-3}{2} = -\frac{3}{2}
\]

### Step 2: Calculate the Y-Intercept (b)

Now that we have the slope \( m \), we can use one of the points to find the y-intercept \( b \). We can use the point \((-2, 7)\).

Plugging the point into the equation \( y = mx + b \):

\[
7 = -\frac{3}{2}(-2) + b
\]

Calculating this:

\[
7 = 3 + b
\]

Now solve for \( b \):

\[
b = 7 - 3 = 4
\]

### Step 3: Write the Function

Now we have both \( m \) and \( b \). The linear function is:

\[
y = -\frac{3}{2}x + 4
\]

### Final Answer

The function that represents the relationship is:

\[
y = -\frac{3}{2}x + 4
\]