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.