Question

if (f(x)) is a linear function, (f(-1) = -2), and (f(5) = 1), find an equation for (f(x)).

(f(x) = \frac{3}{7}x - \frac{11}{7})

Explanation:

⚡ Using what you learned: slope-intercept and standard form

Step 1: Identify the points

The function values \( f(-1) = -2 \) and \( f(5) = 1 \) correspond to the coordinate points:

$$ (-1, -2) \quad \text{and} \quad (5, 1) $$

Step 2: Calculate the slope

Use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):

$$ m = \frac{1 - (-2)}{5 - (-1)} = \frac{1 + 2}{5 + 1} = \frac{3}{6} = \frac{1}{2} $$

Step 3: Find the y-intercept

Use the point-slope form \( y - y_1 = m(x - x_1) \) with the point \( (5, 1) \):

$$ y - 1 = \frac{1}{2}(x - 5) $$

$$ y - 1 = \frac{1}{2}x - \frac{5}{2} $$

$$ y = \frac{1}{2}x - \frac{5}{2} + 1 $$

$$ y = \frac{1}{2}x - \frac{3}{2} $$

Answer:

$$ f(x) = \frac{1}{2}x - \frac{3}{2} $$