Question

1. plot the image of point z under a dilation centered at the origin with scale factor \\(\frac{3}{4}\\). write the coordinates of the image.

1. point a lies on the x-axis. a dilation centered at the origin maps point a to point a. which statements are true? choose all that apply.

a. point a lies on the x-axis.
b. point a lies on the y-axis.
c. point a lies in quadrant i.
d. point a has an x-coordinate of 0.
e. point a has a y-coordinate of 0.

Explanation:

Question 3

Step1: Identify coordinates of Z

From the graph, point Z is at \((0, -8)\) (since it's on the y - axis, x - coordinate is 0 and y - coordinate is - 8).

Step2: Apply dilation formula

The formula for dilation centered at the origin with scale factor \(k\) is \((x,y)\to(kx,ky)\). Here, \(k = \frac{3}{4}\) and the original coordinates of Z are \((0,-8)\).
So, the new x - coordinate is \(\frac{3}{4}\times0 = 0\) and the new y - coordinate is \(\frac{3}{4}\times(-8)=-6\).

• For option A: A dilation centered at the origin preserves the collinearity with the origin. Since point A is on the x - axis, the line from the origin to A is the x - axis. After dilation, the image \(A'\) will still lie on this line (the x - axis), so A is true.
• For option B: Since A is on the x - axis, its image after dilation about the origin will not be on the y - axis (unless A is the origin, but even then, the origin is on both axes, but in general, a point on x - axis (non - origin) will not map to y - axis), so B is false.
• For option C: Point A could be on the negative x - axis. If A is on the negative x - axis, then after dilation (with a positive scale factor), \(A'\) will be on the negative x - axis, not in Quadrant I. So C is not always true.
• For option D: If A is not the origin, its x - coordinate is non - zero. After dilation, the x - coordinate of \(A'\) will be \(k\times x_A\) (where \(k\) is the scale factor and \(x_A

eq0\)), so \(x_{A'}
eq0\). So D is false.

• For option E: Since A is on the x - axis, its y - coordinate is 0. After dilation centered at the origin, the y - coordinate of \(A'\) is \(k\times0 = 0\) (where \(k\) is the scale factor of dilation), so E is true.

Answer:

The coordinates of the image of Z are \((0, - 6)\)

Question 4