Bot GPT 4-o mini 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:
\[ y = \frac{4}{3}x - \frac{4}{3} \]
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