Question

Write an equation for the linear function f such that f(1)= 2 and f (3)=-2 with the fórmulas y-y1=m(x-x1) Support grafically

Answer 1

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.

1. **Point \( (1, 2) \)**:
- When \( x = 1 \):
\[
f(1) = -2(1) + 4 = 2
\]

2. **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.