To find the linear function, we can use the information provided:

1. The point (2,1): This gives us a specific point on the graph, meaning $f(2) = 1$.
2. The rate of change: We are told that $y$ increases by 4 when $x$ increases by 1, which means the slope $m = 4$.

A linear function has the form:

$f(x) = mx + b$

Where:

• $m$ is the slope,
• $b$ is the y-intercept (the value of $f(x)$ when $x = 0$).

From the given information:

• The slope $m = 4$,
• The function passes through $(2, 1)$, so we can substitute this point into the equation to find $b$.

Substitute $x = 2$ and $f(2) = 1$ into the equation:

$1 = 4(2) + b$

Simplifying:

$1 = 8 + b$

Solving for $b$:

$b = 1 - 8 = -7$

Thus, the linear function is:

$f(x) = 4x - 7$

This is the linear function that satisfies the given conditions.

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Would you like further details on this solution?

Here are 5 related questions for deeper exploration:

1. How can we interpret the slope of a line in a real-world context?
2. What happens if the slope is negative instead of positive?
3. How would the function change if the point (2,1) were replaced with (3,1)?
4. What is the meaning of the y-intercept in a linear function graph?
5. Can the function represent a nonlinear relationship? If not, why?

Tip: Always remember that the slope $m$ represents the "rate of change" of the function — how much $y$ increases or decreases as $x$ changes.