Question

point d is shown in the coordinate plane. plot the image point for each single transformation of point d. then write the coordinates. a. image a: reflection across the x - axis b. image b: translation 7 units to the right c. image c: rotation 90° counterclockwise around the origin

Explanation:

1. First, assume the coordinates of point \(D\) from the graph:

• Looking at the graph, the coordinates of point \(D\) are \((- 3,-2)\).

1. a. Reflection across the x - axis:

• Step1: Recall the rule for reflection across the x - axis
• The rule for reflecting a point \((x,y)\) across the x - axis is \((x,-y)\).
• Step2: Apply the rule to point \(D(-3,-2)\)
• For \(x=-3\) and \(y = - 2\), the image \(A\) has coordinates \((-3,2)\).

1. b. Translation 7 units to the right:

• Step1: Recall the rule for translation
• The rule for translating a point \((x,y)\) \(h\) units to the right is \((x + h,y)\), where \(h = 7\) in this case.
• Step2: Apply the rule to point \(D(-3,-2)\)
• Substitute \(x=-3\), \(y=-2\), and \(h = 7\) into the rule. The new \(x\) - coordinate is \(x+h=-3 + 7=4\), and the \(y\) - coordinate remains the same. So the image \(B\) has coordinates \((4,-2)\).

1. c. Rotation 90° counter - clockwise around the origin:

• Step1: Recall the rule for 90° counter - clockwise rotation around the origin
• The rule for rotating a point \((x,y)\) 90° counter - clockwise around the origin is \((-y,x)\).
• Step2: Apply the rule to point \(D(-3,-2)\)
• Substitute \(x=-3\) and \(y=-2\) into the rule. The new \(x\) - coordinate is \(-(-2)=2\), and the new \(y\) - coordinate is \(-3\). So the image \(C\) has coordinates \((2,-3)\).

Answer:

a. The coordinates of image \(A\) are \((-3,2)\).
b. The coordinates of image \(B\) are \((4,-2)\).
c. The coordinates of image \(C\) are \((2,-3)\).