Question

question 6
1 pts
what are the coordinates of the image of p(3, -4) under a reflection in the x - axis?
○ (3, 4)
○ (-3, 4)
○ (3, -4)
○ (-4, 3)
question 7
1 pts
what are the coordinates of (2, 3) after a translation 2 units left and then a rotation of 90° in a clockwise direction about (0, 0)?
○ (0, -2)
○ (3, 0)
○ (-3, 0)
○ (0, 3)
question 8
1 pts

Explanation:

Question 6

Step1: Recall reflection over x - axis rule

The rule for reflecting a point \((x,y)\) over the \(x\) - axis is \((x,y)\to(x, - y)\).

Step2: Apply the rule to \(P(3,-4)\)

For the point \(P(3,-4)\), where \(x = 3\) and \(y=-4\). Using the reflection rule, we change the sign of the \(y\) - coordinate. So the new \(y\) - coordinate is \(-(-4)=4\), and the \(x\) - coordinate remains \(3\). So the image of \(P(3,-4)\) after reflection over the \(x\) - axis is \((3,4)\).

Step1: Perform the translation

The translation of a point \((x,y)\) 2 units left means we subtract 2 from the \(x\) - coordinate. For the point \((2,3)\), after translating 2 units left, the new coordinates are \((2 - 2,3)=(0,3)\).

Step2: Recall the rule for 90° clockwise rotation about the origin

The rule for rotating a point \((x,y)\) 90° clockwise about the origin \((0,0)\) is \((x,y)\to(y,-x)\).

Step3: Apply the rotation rule to \((0,3)\)

For the point \((0,3)\), using the rotation rule \((x,y)\to(y,-x)\), we have \(x = 0\) and \(y = 3\). So the new coordinates are \((3,-0)=(3,0)\)? Wait, no, wait. Wait, the rule for 90° clockwise rotation is \((x,y)\to(y,-x)\). So for \((0,3)\), \(x = 0\), \(y = 3\). Then the new \(x\) is \(y=3\), and the new \(y\) is \(-x = 0\)? Wait, no, I made a mistake. Wait, the correct rule for 90° clockwise rotation about the origin is \((x,y)\to(y, - x)\). So if the point is \((0,3)\), then \(x = 0\), \(y = 3\). So the image after rotation is \((3,-0)=(3,0)\)? Wait, no, let's re - derive the rotation rule. A 90° clockwise rotation about the origin: if we have a point \((x,y)\) in the coordinate plane, to rotate it 90° clockwise, we can think of it as moving the point from the first quadrant (if \(x>0,y>0\)) to the fourth quadrant (or other quadrants depending on the original position). The correct transformation matrix for 90° clockwise rotation is \(

$$\begin{pmatrix}0&1\\-1&0\end{pmatrix}$$

\). So when we multiply the vector \(

$$\begin{pmatrix}x\\y\end{pmatrix}$$

\) by this matrix, we get \(

$$\begin{pmatrix}y\\-x\end{pmatrix}$$

\). So for the point \((0,3)\), \(x = 0\), \(y = 3\). So the new point is \((3,-0)=(3,0)\)? Wait, no, \(-x=-0 = 0\), so \((y,-x)=(3,0)\). Wait, but let's check with an example. If we have a point \((1,0)\), rotating 90° clockwise about the origin should give \((0,-1)\). Using the rule \((x,y)\to(y,-x)\), \((1,0)\to(0,-1)\), which is correct. For the point \((0,3)\), applying \((x,y)\to(y,-x)\), we get \((3,-0)=(3,0)\). Wait, but let's check again. Wait, the translation gives us \((0,3)\). Rotating 90° clockwise about the origin: the formula is \((x,y)\) becomes \((y, - x)\). So \(x = 0\), \(y = 3\), so \((3,-0)=(3,0)\).

Answer:

A. \((3,4)\)

Question 7