Find an equation for the family of linear functions such that
f(4) = 1.
(Use the standard coordinate variables x and y. You may use m for the slope and b for the y-intercept as needed.)

To find the equation for the family of linear functions where \( f(4) = 1 \), let's proceed step by step.

### Step 1: Write the general form of a linear function
A linear function can be written in the form:
\[
f(x) = mx + b
\]
where:
- \( m \) is the slope of the line.
- \( b \) is the y-intercept.

### Step 2: Use the given condition \( f(4) = 1 \)
We know that when \( x = 4 \), the function \( f(x) \) equals 1. Therefore:
\[
f(4) = m(4) + b = 1
\]
This simplifies to:
\[
4m + b = 1
\]

### Step 3: Express \( b \) in terms of \( m \)
To express \( b \) in terms of \( m \), solve the equation:
\[
b = 1 - 4m
\]

### Step 4: Substitute back into the general form
Substitute \( b = 1 - 4m \) into the general form of the linear function:
\[
f(x) = mx + (1 - 4m)
\]

### Final Answer:
The equation for the family of linear functions is:
\[
f(x) = mx + 1 - 4m
\]
where \( m \) is the slope and can take any real value.

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

Here are 5 related questions:
1. How do you find the slope of a linear function given two points?
2. What is the significance of the y-intercept in a linear equation?
3. How do you determine if two linear functions are parallel?
4. Can you find the equation of a line given the slope and a point on the line?
5. What is the equation of a line that is perpendicular to a given line?

**Tip:** When dealing with linear functions, always keep in mind that the slope determines the steepness of the line, and the y-intercept indicates where the line crosses the y-axis.

Discover how to find the equation for a family of linear functions where f(4) = 1. This problem involves determining the slope and y-intercept using a given point on the line. Suitable for high school students and those interested in algebraic functions.

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