Find an expression for the function f(x) whose graph is a line passing through the points (−6,−9) and (−2,−1). f(x) = 

To find the expression for the function \( f(x) \), we need to determine the equation of the line passing through the points \((-6, -9)\) and \((-2, -1)\). The general form of a linear equation is:

\[
f(x) = mx + b
\]

where \( m \) is the slope and \( b \) is the y-intercept.

### Step 1: Find the slope (\( m \))

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substitute the given points \((-6, -9)\) and \((-2, -1)\):

\[
m = \frac{-1 - (-9)}{-2 - (-6)} = \frac{-1 + 9}{-2 + 6} = \frac{8}{4} = 2
\]

### Step 2: Find the y-intercept (\( b \))

Now that we know the slope \( m = 2 \), we can use one of the given points to find the y-intercept. Let's use the point \((-2, -1)\) and substitute into the equation \( f(x) = 2x + b \).

\[
-1 = 2(-2) + b
\]

\[
-1 = -4 + b
\]

Solving for \( b \):

\[
b = -1 + 4 = 3
\]

### Step 3: Write the equation

Now that we have the slope \( m = 2 \) and the y-intercept \( b = 3 \), the equation of the line is:

\[
f(x) = 2x + 3
\]

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Would you like to explore any specific steps in more detail?

Here are some related questions:
1. How do you calculate the slope between any two points?
2. How do you find the y-intercept when given a point and the slope?
3. Can this method be applied to non-linear functions?
4. How would the equation change if the slope was negative?
5. What are the properties of lines parallel or perpendicular to this one?

**Tip:** Always double-check your slope calculation when dealing with negative numbers to avoid simple mistakes.

Discover how to find the equation of a linear function passing through the points (-6, -9) and (-2, -1). This solution covers key algebraic concepts such as the slope formula and the slope-intercept form of a linear equation. Suitable for students in Grades 8-10.

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