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

To find the equation of the linear function \( f(x) \), we can use the general form of a linear function:

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

where \( m \) is the slope and \( b \) is the y-intercept. Given the points \( (-4, 3) \) and \( (5, -5) \), we can determine the slope \( m \) using the formula:

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

Substitute the given points into the formula:

\[ m = \frac{-5 - 3}{5 - (-4)} = \frac{-5 - 3}{5 + 4} = \frac{-8}{9} = -\frac{8}{9} \]

Now that we have the slope \( m = -\frac{8}{9} \), we can use one of the points to find the y-intercept \( b \). Let's use the point \( (-4, 3) \):

\[ f(x) = -\frac{8}{9}x + b \]

Substitute \( x = -4 \) and \( f(-4) = 3 \):

\[ 3 = -\frac{8}{9}(-4) + b \]

\[ 3 = \frac{32}{9} + b \]

To isolate \( b \), subtract \( \frac{32}{9} \) from both sides:

\[ 3 - \frac{32}{9} = b \]

Convert 3 to a fraction with a denominator of 9:

\[ 3 = \frac{27}{9} \]

So,

\[ \frac{27}{9} - \frac{32}{9} = b \]

\[ b = \frac{27 - 32}{9} \]

\[ b = -\frac{5}{9} \]

Therefore, the equation of the linear function is:

\[ f(x) = -\frac{8}{9}x - \frac{5}{9} \]

Would you like any further details or have any questions about this solution?

Here are 5 questions you might ask next:
1. How do I find the slope of a line from two points?
2. Can you explain how to convert a decimal to a fraction?
3. What is the significance of the y-intercept in a linear function?
4. How do you graph a linear function given its equation?
5. Can you show me how to solve a system of linear equations?

**Tip:** When solving for the y-intercept \( b \) in a linear equation, always substitute one of the given points into the equation after determining the slope to solve for \( b \) accurately.

Learn how to find the equation of a linear function when given two points (-4, 3) and (5, -5). This problem covers key algebraic concepts, including the slope formula and equation of a line. Suitable for students in Grades 7-9.

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