To find the equation of the linear function \( f(x) \), we will start by determining the slope \( m \) using the two points provided: \( (1, 2) \) and \( (3, -2) \).
Step 1: Calculate the Slope \( m \)
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the given points \( (1, 2) \) and \( (3, -2) \):
- \( (x_1, y_1) = (1, 2) \)
- \( (x_2, y_2) = (3, -2) \)
We get:
\[ m = \frac{-2 - 2}{3 - 1} = \frac{-4}{2} = -2 \]
Step 2: Use the Point-Slope Form
Now that we have the slope \( m = -2 \), we can use the point-slope formula \( y - y_1 = m(x - x_1) \) with one of the points, let's choose \( (1, 2) \):
\[ y - 2 = -2(x - 1) \]
Step 3: Simplify to Slope-Intercept Form
Now, let's simplify the equation:
\[ y - 2 = -2(x - 1) \] \[ y - 2 = -2x + 2 \] \[ y = -2x + 4 \]
Linear Function Equation
Thus, the equation of the linear function is:
\[ f(x) = -2x + 4 \]
Graphical Representation
To graph the function, we can plot the points \( (1, 2) \) and \( (3, -2) \) and draw the line through these points.
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Point \( (1, 2) \):
- When \( x = 1 \): \[ f(1) = -2(1) + 4 = 2 \]
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Point \( (3, -2) \):
- When \( x = 3 \): \[ f(3) = -2(3) + 4 = -2 \]
Now plot these points on a graph and draw a line through them:
- Point \( (1, 2) \) is above the x-axis at x = 1 and y = 2.
- Point \( (3, -2) \) is below the x-axis at x = 3 and y = -2.
Connecting these points gives you the linear function, which slopes downward due to the negative slope of -2.
Conclusion
The equation of the linear function is \( f(x) = -2x + 4 \) and it passes through the points \( (1, 2) \) and \( (3, -2) \). You can visualize it by graphing the line based on the points we've calculated.
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