(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.