Question

lesson 27
understanding the four-quadrant
coordinate plane

for problems 1-6, plot and label each point in the coordinate plane. name the quadrant or axis where the point is located.

1. $a(-3, -2)$
2. $b(4, -4)$
3. $c(2, 3)$
4. $d(-2, 4)$
5. $e(3, -3)$
6. $f(4, 0)$
7. if point $e$ above is reflected across the $x$-axis, what would be the coordinates of the reflection? explain.
8. imagine that one of the points given in problems 1-6 has been reflected. the reflection is in quadrant ii. what are the possible coordinates of the reflected point? explain.
9. bradley says that if point $b$ is reflected across the $y$-axis and its reflection is then reflected across the $x$-axis, the result is point $d$. is bradley correct? explain.

Explanation:

Step1: Identify quadrant rules

A coordinate $(x,y)$ follows:

• Quadrant I: $x>0,y>0$
• Quadrant II: $x<0,y>0$
• Quadrant III: $x<0,y<0$
• Quadrant IV: $x>0,y<0$
• $x$-axis: $y=0$; $y$-axis: $x=0$

Step2: Analyze point $A(-3,-2)$

$x=-3<0$, $y=-2<0$ → Quadrant III

Step3: Analyze point $B(4,-4)$

$x=4>0$, $y=-4<0$ → Quadrant IV

Step4: Analyze point $C(2,3)$

$x=2>0$, $y=3>0$ → Quadrant I

Step5: Analyze point $D(-2,4)$

$x=-2<0$, $y=4>0$ → Quadrant II

Step6: Analyze point $E(3,-3)$

$x=3>0$, $y=-3<0$ → Quadrant IV

Step7: Analyze point $F(4,0)$

$y=0$ → $x$-axis

Step8: Reflect $E$ over $x$-axis

Rule: $(x,y)\to(x,-y)$. $E(3,-3)\to(3, 3)$

Step9: Find pre-images for Quadrant II reflection

Quadrant II: $x<0,y>0$. Possible reflections:

• From Quadrant I: $(x,y)\to(-x,y)$ → pre-image $(a,b),a>0,b>0$ (point $C(2,3)$ → $(-2,3)$)
• From Quadrant III: $(x,y)\to(x,-y)$ → pre-image $(a,b),a<0,b<0$ (point $A(-3,-2)$ → $(-3,2)$)
• From Quadrant IV: $(x,y)\to(-x,-y)$ → pre-image $(a,b),a>0,b<0$ (points $B(4,-4),E(3,-3)$ → $(-4,4),(-3,3)$)
• From $x$-axis: $(x,0)\to(-x,0)$ (not Quadrant II, discard)

Step10: Verify Bradley's claim

1. Reflect $B(4,-4)$ over $y$-axis: $(x,y)\to(-x,y)$ → $(-4,-4)$
2. Reflect $(-4,-4)$ over $x$-axis: $(x,y)\to(x,-y)$ → $(-4,4)$

Point $D$ is $(-2,4)$, not $(-4,4)$ → Bradley is wrong.

Answer:

1. $A(-3,-2)$: Quadrant III
2. $B(4,-4)$: Quadrant IV
3. $C(2,3)$: Quadrant I
4. $D(-2,4)$: Quadrant II
5. $E(3,-3)$: Quadrant IV
6. $F(4,0)$: $x$-axis
7. Reflected $E$: $(3, 3)$; reflection over $x$-axis flips the $y$-sign.
8. Possible coordinates: $(-2,3), (-3,2), (-4,4), (-3,3)$; these come from reflecting Quadrant I/III/IV points over $y$-axis, $x$-axis, or origin to reach Quadrant II.
9. Bradley is incorrect; reflecting $B$ over $y$-axis then $x$-axis gives $(-4,4)$, not $D(-2,4)$.