Question

1. use the coordinate plane to answer the questions:

a. the point (-6,-2) is located in quadrant
b. if the point (2,-4) is reflected over the x - axis, then it will be in quadrant
c. label point a at (5,-1) and point b at (-7,-4).
d. how is (5,9) different than (-5,9)?

Explanation:

Step1: Determine the quadrant of \((-6,-2)\)

In the coordinate plane, quadrant III has negative \(x\) - values and negative \(y\) - values. For the point \((-6,-2)\), since \(x=-6<0\) and \(y = - 2<0\), it is in quadrant III.

Step2: Find the reflection of \((2,-4)\) over the \(x\) - axis

The rule for reflecting a point \((x,y)\) over the \(x\) - axis is \((x,-y)\). For the point \((2,-4)\), after reflection over the \(x\) - axis, the new point is \((2,4)\). Since \(x = 2>0\) and \(y=4>0\), it is in quadrant I.

Step3: Analyze the difference between \((5,9)\) and \((-5,9)\)

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \(x_1 = 5,y_1=9,x_2=-5,y_2 = 9\). Then \(d=\sqrt{(-5 - 5)^2+(9 - 9)^2}=\sqrt{(-10)^2+0^2}=10\). The \(y\) - values are the same (\(y = 9\)), and the \(x\) - values are additive inverses. The two points are symmetric about the \(y\) - axis.

Answer:

a. III; b. I; d. The two points are symmetric about the \(y\) - axis (or the distance between them is \(10\) units along the \(x\) - direction while \(y\) - values are equal)