Question

The graph of a linear function is shown on the grid.
It passes through 4, 4 and -2, -4.

Which equation is best represented by this graph?

Answer 1

To find the equation of the linear function that passes through the points (4, 4) and (-2, -4), we can use the slope-intercept form of a linear equation, which is \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.

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

The slope \( m \) is calculated using the formula:

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

Using the points (4, 4) as \((x_1, y_1)\) and (-2, -4) as \((x_2, y_2)\):

\[
m = \frac{-4 - 4}{-2 - 4} = \frac{-8}{-6} = \frac{4}{3}
\]

### Step 2: Use the slope and a point to find the y-intercept (b)

Now we can use one of the points and the slope to find \( b \). We'll use the point (4, 4):

\[
y = mx + b
\]
\[
4 = \frac{4}{3}(4) + b
\]
\[
4 = \frac{16}{3} + b
\]

To solve for \( b \), we first convert 4 to a fraction with a denominator of 3:

\[
4 = \frac{12}{3}
\]
\[
\frac{12}{3} = \frac{16}{3} + b
\]

Now isolate \( b \):

\[
b = \frac{12}{3} - \frac{16}{3} = \frac{-4}{3}
\]

### Step 3: Write the equation

Now, we have \( m = \frac{4}{3} \) and \( b = -\frac{4}{3} \). Plugging these values into the slope-intercept equation gives us:

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

### Conclusion

The equation that best represents the graph of the linear function that passes through the points (4, 4) and (-2, -4) is:

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