Question

4.2 assignment
in exercises 1-4, graph the point or polygon and its image after a reflection in the given line
1 j(2, -4), k(3, 7), l(6, -1); x-axis
2 j(5, 3); y-axis
3 j(1, -1), k(3, 0), l(0, -4); x = 2
4 j(3, -5); y = -3

in exercises 5 and 6, state the new image after a reflection in the line y = x.

1. a(0, -3), b(2, 2), c(5, 0)
2. a(2, -1), b(-1, 2), c(2, 3), d(4, 2)

(handwritten answers: 5. a(-3,0), b(2,2), c(0,5); 6. a(-1,2), b(2,-1), c(3,2), d(2,4))

Explanation:

Let's solve these reflection problems step by step. We'll use the rules for reflecting points over different lines (x - axis, y - axis, \(x = a\), \(y = b\), and \(y=x\)).

Exercise 1: Reflect \(J(2,-4)\), \(K(3,7)\), \(L(6,-1)\) over the \(x\) - axis

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

• For \(J(2,-4)\):
• Apply the rule: \(x = 2\), \(y=-4\), so the image \(J'\) is \((2,4)\) (since \(-(-4)=4\)).
• For \(K(3,7)\):
• Apply the rule: \(x = 3\), \(y = 7\), so the image \(K'\) is \((3,-7)\) (since \(-7=-7\)).
• For \(L(6,-1)\):
• Apply the rule: \(x = 6\), \(y=-1\), so the image \(L'\) is \((6,1)\) (since \(-(-1)=1\)).

Exercise 2: Reflect \(J(5,3)\) over the \(y\) - axis

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

• For \(J(5,3)\):
• Apply the rule: \(x = 5\), so \(-x=-5\), and \(y = 3\). The image \(J'\) is \((-5,3)\).

Exercise 3: Reflect \(J(1,-1)\), \(K(3,0)\), \(L(0,-4)\) over the line \(x = 2\)

The rule for reflecting a point \((x,y)\) over the vertical line \(x=a\) is \((x,y)\to(2a - x,y)\). Here \(a = 2\), so the formula becomes \((x,y)\to(4 - x,y)\).

• For \(J(1,-1)\):
• Substitute \(x = 1\) into \(4 - x\): \(4-1 = 3\), \(y=-1\). So \(J'\) is \((3,-1)\).
• For \(K(3,0)\):
• Substitute \(x = 3\) into \(4 - x\): \(4 - 3=1\), \(y = 0\). So \(K'\) is \((1,0)\).
• For \(L(0,-4)\):
• Substitute \(x = 0\) into \(4 - x\): \(4-0 = 4\), \(y=-4\). So \(L'\) is \((4,-4)\).

Exercise 4: Reflect \(J(3,-5)\) over the line \(y=-3\)

The rule for reflecting a point \((x,y)\) over the horizontal line \(y = b\) is \((x,y)\to(x,2b - y)\). Here \(b=-3\), so the formula becomes \((x,y)\to(x,-6 - y)\).

• For \(J(3,-5)\):
• Substitute \(y=-5\) into \(-6 - y\): \(-6-(-5)=-6 + 5=-1\), \(x = 3\). So \(J'\) is \((3,-1)\).

Exercise 5: Reflect \(A(0,-3)\), \(B(2,2)\), \(C(5,0)\) over the line \(y = x\)

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

• For \(A(0,-3)\):
• Swap \(x\) and \(y\): \(A'\) is \((-3,0)\).
• For \(B(2,2)\):
• Swap \(x\) and \(y\): \(B'\) is \((2,2)\) (since \(x=y = 2\)).
• For \(C(5,0)\):
• Swap \(x\) and \(y\): \(C'\) is \((0,5)\).

Exercise 6: Reflect \(A(2,-1)\), \(B(-1,2)\), \(C(2,3)\), \(D(4,2)\) over the line \(y = x\)

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

• For \(A(2,-1)\):
• Swap \(x\) and \(y\): \(A'\) is \((-1,2)\).
• For \(B(-1,2)\):
• Swap \(x\) and \(y\): \(B'\) is \((2,-1)\).
• For \(C(2,3)\):
• Swap \(x\) and \(y\): \(C'\) is \((3,2)\).
• For \(D(4,2)\):
• Swap \(x\) and \(y\): \(D'\) is \((2,4)\).

Final Answers (Summarized)

1. \(J'(2,4)\), \(K'(3,-7)\), \(L'(6,1)\)
2. \(J'(-5,3)\)
3. \(J'(3,-1)\), \(K'(1,0)\), \(L'(4,-4)\)
4. \(J'(3,-1)\)
5. \(A'(-3,0)\), \(B'(2,2)\), \(C'(0,5)\)
6. \(A'(-1,2)\), \(B'(2,-1)\), \(C'(3,2)\), \(D'(2,4)\)

Answer:

Let's solve these reflection problems step by step. We'll use the rules for reflecting points over different lines (x - axis, y - axis, \(x = a\), \(y = b\), and \(y=x\)).

Exercise 1: Reflect \(J(2,-4)\), \(K(3,7)\), \(L(6,-1)\) over the \(x\) - axis

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

• For \(J(2,-4)\):
• Apply the rule: \(x = 2\), \(y=-4\), so the image \(J'\) is \((2,4)\) (since \(-(-4)=4\)).
• For \(K(3,7)\):
• Apply the rule: \(x = 3\), \(y = 7\), so the image \(K'\) is \((3,-7)\) (since \(-7=-7\)).
• For \(L(6,-1)\):
• Apply the rule: \(x = 6\), \(y=-1\), so the image \(L'\) is \((6,1)\) (since \(-(-1)=1\)).

Exercise 2: Reflect \(J(5,3)\) over the \(y\) - axis

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

• For \(J(5,3)\):
• Apply the rule: \(x = 5\), so \(-x=-5\), and \(y = 3\). The image \(J'\) is \((-5,3)\).

Exercise 3: Reflect \(J(1,-1)\), \(K(3,0)\), \(L(0,-4)\) over the line \(x = 2\)

The rule for reflecting a point \((x,y)\) over the vertical line \(x=a\) is \((x,y)\to(2a - x,y)\). Here \(a = 2\), so the formula becomes \((x,y)\to(4 - x,y)\).

• For \(J(1,-1)\):
• Substitute \(x = 1\) into \(4 - x\): \(4-1 = 3\), \(y=-1\). So \(J'\) is \((3,-1)\).
• For \(K(3,0)\):
• Substitute \(x = 3\) into \(4 - x\): \(4 - 3=1\), \(y = 0\). So \(K'\) is \((1,0)\).
• For \(L(0,-4)\):
• Substitute \(x = 0\) into \(4 - x\): \(4-0 = 4\), \(y=-4\). So \(L'\) is \((4,-4)\).

Exercise 4: Reflect \(J(3,-5)\) over the line \(y=-3\)

The rule for reflecting a point \((x,y)\) over the horizontal line \(y = b\) is \((x,y)\to(x,2b - y)\). Here \(b=-3\), so the formula becomes \((x,y)\to(x,-6 - y)\).

• For \(J(3,-5)\):
• Substitute \(y=-5\) into \(-6 - y\): \(-6-(-5)=-6 + 5=-1\), \(x = 3\). So \(J'\) is \((3,-1)\).

Exercise 5: Reflect \(A(0,-3)\), \(B(2,2)\), \(C(5,0)\) over the line \(y = x\)

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

• For \(A(0,-3)\):
• Swap \(x\) and \(y\): \(A'\) is \((-3,0)\).
• For \(B(2,2)\):
• Swap \(x\) and \(y\): \(B'\) is \((2,2)\) (since \(x=y = 2\)).
• For \(C(5,0)\):
• Swap \(x\) and \(y\): \(C'\) is \((0,5)\).

Exercise 6: Reflect \(A(2,-1)\), \(B(-1,2)\), \(C(2,3)\), \(D(4,2)\) over the line \(y = x\)

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

• For \(A(2,-1)\):
• Swap \(x\) and \(y\): \(A'\) is \((-1,2)\).
• For \(B(-1,2)\):
• Swap \(x\) and \(y\): \(B'\) is \((2,-1)\).
• For \(C(2,3)\):
• Swap \(x\) and \(y\): \(C'\) is \((3,2)\).
• For \(D(4,2)\):
• Swap \(x\) and \(y\): \(D'\) is \((2,4)\).

Final Answers (Summarized)

1. \(J'(2,4)\), \(K'(3,-7)\), \(L'(6,1)\)
2. \(J'(-5,3)\)
3. \(J'(3,-1)\), \(K'(1,0)\), \(L'(4,-4)\)
4. \(J'(3,-1)\)
5. \(A'(-3,0)\), \(B'(2,2)\), \(C'(0,5)\)
6. \(A'(-1,2)\), \(B'(2,-1)\), \(C'(3,2)\), \(D'(2,4)\)