Find an equation for the family of linear functions with slope 4. (Use the standard coordinate variables x and y. You may use m for the slope and b for the y-intercept as needed.) y= 4x+b Sketch several members of the family. y 10 10 + -10 5 10 -10 -5 o y 101 51 x -10 -5 5 10 -10 -5 5 10 O -10 (b) 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.) y=m(n-6)+1 Sketch several members of the family. y 10F у 101 5! 10 -10 -101 y 10H y 10H X -10 -5 -10 -5 -5H -5 -101 -101 (c) What equation belongs to both families? (Use the standard coordinate variables x and y.) y=mx + b

### (a) Family of Linear Functions with Slope 4:

The equation for a family of linear functions where the slope \( m = 4 \) is given by:

\[
y = 4x + b
\]

Here, \( b \) represents the y-intercept, which can vary. By changing the value of \( b \), we get different members of the family of functions. For example:
- \( y = 4x + 0 \)
- \( y = 4x + 2 \)
- \( y = 4x - 3 \)

### (b) Family of Linear Functions where \( f(4) = 1 \):

We need to find the equation for a family of linear functions such that when \( x = 4 \), the value of the function is 1 (i.e., \( f(4) = 1 \)).

The general form of a linear equation is:

\[
y = mx + b
\]

Given \( f(4) = 1 \), substitute \( x = 4 \) and \( y = 1 \):

\[
1 = m(4) + b
\]

Solve for \( b \):

\[
b = 1 - 4m
\]

Thus, the equation for the family of linear functions is:

\[
y = mx + (1 - 4m)
\]

### (c) What Equation Belongs to Both Families?

We want to find an equation that belongs to both families. From part (a), the slope is 4. Substituting \( m = 4 \) into the equation from part (b):

\[
y = 4x + (1 - 4 \cdot 4)
\]
\[
y = 4x + (1 - 16)
\]
\[
y = 4x - 15
\]

Thus, the equation that belongs to both families is:

\[
y = 4x - 15
\]

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Would you like more details or have any questions on the solution?

Here are 5 related questions to explore the topic further:
1. How do you determine the slope from a given linear equation?
2. How does the value of \( b \) in the equation \( y = mx + b \) affect the graph of the function?
3. What happens to the graph if the slope changes but the y-intercept remains constant?
4. Can you find the equation of a line passing through two given points?
5. How would you solve for \( x \) if a specific \( y \)-value is given?

**Tip**: When graphing a family of linear functions, focus on how the slope (steepness) and y-intercept (vertical shift) affect the overall behavior of the graph.

Find an equation for the family of linear functions with slope 4. (Use the standard coordinate variables x and y. You may use m for the slope and b for the y-intercept as needed.) y= 4x+b Sketch several members of the family. (b) 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.) y=m(n-6)+1 Sketch several members of the family. (c) What equation belongs to both families? (Use the standard coordinate variables x and y.) y=mx + b

Learn how to solve problems involving families of linear functions. This example covers finding the general equation for a family of linear functions with slope 4, as well as a family where f(4) = 1. Suitable for students in Grades 8-10, this guide includes key concepts such as the slope-intercept form and linear graph transformations.

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