if a function has a rate of change 1/2 and y=4 when x=2, write its equation

To write the equation of the function, we can use the point-slope form of a linear equation. The point-slope form is given by:

\[
y - y_1 = m(x - x_1)
\]

where:
- \(m\) is the slope (rate of change),
- \((x_1, y_1)\) is a point on the line.

Given:
- Rate of change (slope) \(m = \frac{1}{2}\),
- Point \((x_1, y_1) = (2, 4)\).

Substituting these values into the point-slope form:

\[
y - 4 = \frac{1}{2}(x - 2)
\]

Now, simplify the equation:

\[
y - 4 = \frac{1}{2}x - 1
\]

Adding 4 to both sides to solve for \(y\):

\[
y = \frac{1}{2}x + 3
\]

So, the equation of the function is:

\[
y = \frac{1}{2}x + 3
\]

Would you like more details or have any questions? Here are some related questions to explore:

1. How would the equation change if the slope were \(\frac{3}{4}\) instead of \(\frac{1}{2}\)?
2. What is the y-intercept of the function \(y = \frac{1}{2}x + 3\)?
3. How can you find the x-intercept of the function \(y = \frac{1}{2}x + 3\)?
4. What does the slope of \(\frac{1}{2}\) tell you about the function's graph?
5. How do you convert the point-slope form to the slope-intercept form?

**Tip:** Always check if the equation of the line correctly satisfies the given point by substituting the values of \(x\) and \(y\) back into the final equation.

Learn how to find the equation of a line with a slope of 1/2 that passes through the point (2, 4). This problem covers essential algebraic concepts, including point-slope and slope-intercept forms, and is suitable for students in Grades 7-9.

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