If f(x) is a linear function, f(-2)=-3, and f(4)=-5, find an equation for f(x)

f(x) =  

To find the equation for the linear function \( f(x) \), we need to determine the slope \( m \) and the y-intercept \( b \) of the line. The general form of a linear equation is:

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

### Step 1: Find the slope \( m \)
The slope \( m \) is calculated as the change in \( f(x) \) over the change in \( x \). We are given two points: 
\[
(-2, -3) \quad \text{and} \quad (4, -5)
\]
Using the slope formula:
\[
m = \frac{f(4) - f(-2)}{4 - (-2)} = \frac{-5 - (-3)}{4 - (-2)} = \frac{-5 + 3}{4 + 2} = \frac{-2}{6} = -\frac{1}{3}
\]

### Step 2: Find the y-intercept \( b \)
Now that we know \( m = -\frac{1}{3} \), we can use one of the given points to solve for \( b \). Let's use the point \( (-2, -3) \).

Substitute \( x = -2 \), \( f(x) = -3 \), and \( m = -\frac{1}{3} \) into the equation:
\[
-3 = -\frac{1}{3}(-2) + b
\]
Simplify:
\[
-3 = \frac{2}{3} + b
\]
Subtract \( \frac{2}{3} \) from both sides:
\[
b = -3 - \frac{2}{3} = \frac{-9}{3} - \frac{2}{3} = \frac{-11}{3}
\]

### Step 3: Write the equation
Now that we know \( m = -\frac{1}{3} \) and \( b = \frac{-11}{3} \), the equation for \( f(x) \) is:
\[
f(x) = -\frac{1}{3}x - \frac{11}{3}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you verify the correctness of the linear equation using given points?
2. What are the steps for finding the slope between two points?
3. How do you graph a linear function given the equation?
4. How would the equation change if the slope were positive?
5. What is the significance of the y-intercept in a linear equation?

**Tip**: Always double-check your slope calculation by ensuring you subtract the values of \( f(x) \) and \( x \) in the correct order!

If f(x) is a linear function, f(-2)=-3, and f(4)=-5, find an equation for f(x)

Learn how to find the equation of a linear function given two points: f(-2) = -3 and f(4) = -5. This problem includes a step-by-step solution using the slope formula and solving for the y-intercept. Perfect for students in Grades 8-10, covering algebraic concepts such as slopes and linear equations.

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