Question

question 13
1 pts
what is the image of (-2,3) after a rotation of 180° counterclockwise?
○ (-3, -2)
○ (2, -3)
○ (2, 3)
○ (3, -2)
question 14
1 pts
if a point in quadrant iii is reflected in the x-axis, its image will lie in quadrant ____.
○ iv
○ on the y-axis
○ ii
○ i

Explanation:

Question 13

Step1: Recall 180° rotation rule

For a point \((x, y)\), a \(180^\circ\) counterclockwise (or clockwise) rotation has the rule \((x, y) \to (-x, -y)\).

Step2: Apply the rule to \((-2, 3)\)

Here, \(x = -2\) and \(y = 3\). So, \(-x = -(-2)=2\) and \(-y = -3\). Thus, the image is \((2, -3)\).

A point in Quadrant III has coordinates \((-, -)\) (negative \(x\), negative \(y\)). Reflecting over the \(x\)-axis changes the \(y\)-coordinate's sign: \((x, y) \to (x, -y)\). So a \((-, -)\) point becomes \((-, +)\), which is Quadrant II? Wait, no—wait, Quadrant III: \((-x, -y)\) ( \(x>0\), \(y>0\) ). Reflect over \(x\)-axis: \((-x, y)\). Wait, no: reflection over \(x\)-axis is \((x, y) \to (x, -y)\). So if original is \((-a, -b)\) ( \(a>0\), \(b>0\) ), after reflection: \((-a, b)\). Which is Quadrant II? Wait, no, Quadrant II is \((-, +)\), Quadrant IV is \((+, -)\). Wait, no, let's correct: Quadrant I: \((+, +)\), II: \((-, +)\), III: \((-, -)\), IV: \((+, -)\). So a point in III: \((-, -)\). Reflect over \(x\)-axis: \(y\) becomes positive, so \((-, +)\), which is Quadrant II? Wait, but the options have II as an option. Wait, no, wait the options: the options are IV, on y-axis, II, I. Wait, maybe I made a mistake. Wait, no: reflection over \(x\)-axis: \((x, y) \to (x, -y)\). So if original is \((-3, -4)\) (III), after reflection: \((-3, 4)\) (II). So the image is in Quadrant II. Wait, but the options have "II" as an option. Wait, but let's check again. Wait, maybe the question was misread. Wait, the question says "a point in Quadrant III is reflected in the x - axis". So coordinates \((x, y)\) with \(x<0\), \(y<0\). After reflection: \((x, -y)\). Since \(y<0\), \(-y>0\). So \(x<0\), \(y>0\): that's Quadrant II. So the answer is II.

Answer:

B. \((2, -3)\)

Question 14