Question

mp structure the coordinates of a point and its image after a reflection are given. identify the line of reflection.

1. (2, - 2)→(2, 2)
2. (-4, 1)→(4, 1)
3. (-2, -5)→(4, -5)
4. (-3, -4)→(-3, 0)

Explanation:

Step1: Recall the rules of reflection

For a reflection over a vertical line \(x = a\), the formula for the reflection of a point \((x,y)\) is \((2a - x,y)\). For a reflection over a horizontal line \(y = b\), the formula is \((x,2b - y)\).

Step2: Analyze problem 27

Given the point \((2,-2)\) and its image \((2,2)\). The \(x -\)coordinate remains the same (\(x = 2\)), and the \(y -\)coordinate changes. Using the formula for reflection over a horizontal line \(y=b\), we have \(y_1=-2\) and \(y_2 = 2\). Substituting into \(y_2=2b - y_1\), we get \(2 = 2b-(-2)\). Solving for \(b\):

$$
LATEXBLOCK0
$$

The line of reflection is \(y = 0\) (the \(x -\)axis).

Step3: Analyze problem 28

Given the point \((-4,1)\) and its image \((4,1)\). The \(y -\)coordinate remains the same (\(y = 1\)), and the \(x -\)coordinate changes. Using the formula for reflection over a vertical line \(x=a\), we have \(x_1=-4\) and \(x_2 = 4\). Substituting into \(x_2=2a - x_1\), we get \(4=2a-(-4)\). Solving for \(a\):

$$
LATEXBLOCK1
$$

The line of reflection is \(x = 0\) (the \(y -\)axis).

Step4: Analyze problem 29

Given the point \((-2,-5)\) and its image \((4,-5)\). The \(y -\)coordinate remains the same (\(y=-5\)), and the \(x -\)coordinate changes. Using the formula for reflection over a vertical line \(x=a\), we have \(x_1=-2\) and \(x_2 = 4\). Substituting into \(x_2=2a - x_1\), we get \(4=2a-(-2)\). Solving for \(a\):

$$
LATEXBLOCK2
$$

The line of reflection is \(x = 1\).

Step5: Analyze problem 30

Given the point \((-3,-4)\) and its image \((-3,0)\). The \(x -\)coordinate remains the same (\(x=-3\)), and the \(y -\)coordinate changes. Using the formula for reflection over a horizontal line \(y=b\), we have \(y_1=-4\) and \(y_2 = 0\). Substituting into \(y_2=2b - y_1\), we get \(0=2b-(-4)\). Solving for \(b\):

$$
LATEXBLOCK3
$$

The line of reflection is \(y=-2\).

Answer:

1. The line of reflection is \(y = 0\) (the \(x -\)axis).
2. The line of reflection is \(x = 0\) (the \(y -\)axis).
3. The line of reflection is \(x = 1\).
4. The line of reflection is \(y=-2\).