To reflect the point $(-2, 2)$ over the line $y = \frac{3}{4}x + 3$, we'll follow these steps:

1. Find the slope of the perpendicular line:
   The slope of the given line $y = \frac{3}{4}x + 3$ is $\frac{3}{4}$.
   The slope of the line perpendicular to it is the negative reciprocal, which is $-\frac{4}{3}$.

2. Write the equation of the perpendicular line:
   The equation of the perpendicular line passing through the point $(-2, 2)$ will be in the form $y - y_1 = m(x - x_1)$, where $m = -\frac{4}{3}$ is the slope and $(-2, 2)$ is the point.
   So, we substitute: $y - 2 = -\frac{4}{3}(x + 2)$ Simplify the equation to get the equation of the perpendicular line: $y = -\frac{4}{3}x - \frac{8}{3} + 2$ $y = -\frac{4}{3}x + \frac{2}{3}$

3. Find the intersection of the two lines:
   Now, we solve the system of equations to find the intersection of the line $y = \frac{3}{4}x + 3$ and the perpendicular line $y = -\frac{4}{3}x + \frac{2}{3}$.

   Set the equations equal to each other: $\frac{3}{4}x + 3 = -\frac{4}{3}x + \frac{2}{3}$ Multiply through by 12 to eliminate the fractions: $9x + 36 = -16x + 8$ Combine like terms: $25x = -28$ So, $x = -\frac{28}{25}$.

4. Find the corresponding y-coordinate:
   Substitute $x = -\frac{28}{25}$ into the equation of the original line $y = \frac{3}{4}x + 3$: $y = \frac{3}{4} \times -\frac{28}{25} + 3 = -\frac{84}{100} + 3 = -0.84 + 3 = 2.16$ So, the point of intersection is $\left(-\frac{28}{25}, 2.16\right)$.

5. Find the reflected point:
   The point $(-2, 2)$ and its reflection will be equidistant from the line. Using the formula for reflection, the reflected point will be $(-3.12, 2.32)$.

The reflection of the point $(-2, 2)$ over the line $y = \frac{3}{4}x + 3$ is approximately $(-3.12, 2.32)$.

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Would you like further details or clarification on any step?

Here are 5 related questions to consider:

1. How do you find the equation of a line given two points?
2. What is the geometric meaning of a line's slope?
3. How would the process change if the reflection line was vertical or horizontal?
4. How can you verify if two lines are perpendicular?
5. What is the formula for the distance between a point and a line?

Tip: Always verify by checking if the midpoint between the original and reflected point lies on the reflection line!