Consider the following: f(-4)=-5, f(1)=4. Write the linear function f that has the given function values.

To find the linear function \( f(x) \) based on the given values \( f(-4) = -5 \) and \( f(1) = 4 \), we can use the point-slope form of a linear equation. A linear function can be written as:

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

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

### Step 1: Calculate the slope
We know two points on the line: \( (-4, -5) \) and \( (1, 4) \). The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

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

Substitute the given points:

\[
m = \frac{4 - (-5)}{1 - (-4)} = \frac{4 + 5}{1 + 4} = \frac{9}{5}
\]

So, the slope is \( m = \frac{9}{5} \).

### Step 2: Find the y-intercept
We can now use the slope-intercept form \( f(x) = mx + b \) and substitute one of the points to solve for \( b \). Using the point \( (1, 4) \):

\[
4 = \frac{9}{5}(1) + b
\]

Solve for \( b \):

\[
4 = \frac{9}{5} + b
\]

Multiply both sides by 5 to eliminate the fraction:

\[
20 = 9 + 5b
\]

Subtract 9 from both sides:

\[
11 = 5b
\]

Now, solve for \( b \):

\[
b = \frac{11}{5}
\]

### Step 3: Write the equation
Now that we have the slope \( m = \frac{9}{5} \) and the y-intercept \( b = \frac{11}{5} \), the linear function is:

\[
f(x) = \frac{9}{5}x + \frac{11}{5}
\]

### Final Answer:
\[
f(x) = \frac{9}{5}x + \frac{11}{5}
\]

Would you like further details on any of the steps?

Here are some related questions to further your understanding:

1. How do you convert a linear equation from point-slope form to slope-intercept form?
2. What is the significance of the slope in a linear function?
3. How can you graph a linear equation using only two points?
4. How do you determine the x-intercept of a linear function?
5. How would the function change if we modified the slope or the y-intercept?

**Tip**: When given two points, always calculate the slope first as it's the key to forming the linear equation.

Discover how to write a linear function f(x) based on given values f(-4) = -5 and f(1) = 4. This step-by-step guide covers key algebraic concepts, including slope calculation and using the slope-intercept form. Suitable for students in Grades 8-10.

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