Question

find a linear function f, given \\(f(12) = -2\\) and \\(f(-12) = -10\\). then find \\(f(0)\\).

\\(f(x) = \\)
(type an expression using x as the variable. simplify your answer.)

Explanation:

Identify the given points

The problem gives two function values:
\(f(12) = -2\) and \(f(-12) = -10\).
These correspond to two coordinate points:
\((x_1, y_1) = (12, -2)\)
\((x_2, y_2) = (-12, -10)\)

Calculate the slope

Using the Slope from Two Points formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Substitute the coordinates:

$$m = \frac{-10 - (-2)}{-12 - 12} = \frac{-8}{-24} = \frac{1}{3}$$

Write the linear function

Using the Point-Slope Form with point \((12, -2)\):

$$y - y_1 = m(x - x_1)$$

$$y - (-2) = \frac{1}{3}(x - 12)$$

$$y + 2 = \frac{1}{3}x - 4$$

$$f(x) = \frac{1}{3}x - 6$$

Find the value of f(0)

Substitute \(x = 0\) into the function:

$$f(0) = \frac{1}{3}(0) - 6 = -6$$

Answer:

Question 1

\(f(x) =\) <blank>\(\frac{1}{3}x - 6\)</blank>

Question 2

\(f(0) =\) <blank>\(-6\)</blank>