Two-dimensional Shapes>Explaining Effects of Translations, Reflections, and Rotations Using Algebraic Representations(TEKS.Math.8.10.C)
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Texas 8th Grade Math › Two-dimensional Shapes>Explaining Effects of Translations, Reflections, and Rotations Using Algebraic Representations(TEKS.Math.8.10.C)
Reflect $P(3,-2)$ over the $x$-axis. What are the new coordinates?
$(-3,2)$
$(3,2)$
$(3,-2)$
$(-3,-2)$
Explanation
Reflection over the $x$-axis uses the rule $(x,y)\to(x,-y)$. Substitute $P(3,-2)$: $(3,-(-2))=(3,2)$. Geometrically, reflecting across the $x$-axis keeps $x$ the same and flips the sign of $y$, moving the point straight up or down.
Rotate $A(2,5)$ $90^\circ$ clockwise about the origin. What are the new coordinates?
$(-5,2)$
$(-2,-5)$
$(-5,-2)$
$(5,-2)$
Explanation
A $90^\circ$ clockwise rotation uses $(x,y)\to(y,-x)$. Substitute $(2,5)$: $(5,-2)$. Geometrically, the coordinates swap and the original $x$ becomes negative.
Points $Q(-4,1)\to Q'(4,1)$ and $R(3,-5)\to R'(-3,-5)$. Which rule describes this transformation?
$(x,y)\to(-x,y)$
$(x,y)\to(x,-y)$
$(x,y)\to(y,x)$
$(x,y)\to(-x,-y)$
Explanation
Both images change the sign of $x$ while $y$ stays the same, so the rule is $(x,y)\to(-x,y)$. For example, $(-4,1)\to(4,1)$ and $(3,-5)\to(-3,-5)$. This is a reflection across the $y$-axis.
Rotate $B(-6,2)$ $180^\circ$ about the origin. What are the new coordinates?
$(-6,-2)$
$(6,2)$
$(6,-2)$
$(2,-6)$
Explanation
A $180^\circ$ rotation uses $(x,y)\to(-x,-y)$. Substitute $(-6,2)$: $(6,-2)$. Geometrically, a $180^\circ$ rotation sends a point to the opposite side through the origin, changing both signs.
Point $M(-1,7)$ is translated by the rule $(x,y)\to(x+4,y-2)$. What are the new coordinates?
$(-5,5)$
$(3,5)$
$(3,-5)$
$(-3,9)$
Explanation
Apply $(x,y)\to(x+4,y-2)$: $x=-1\to-1+4=3$ and $y=7\to7-2=5$, so $(3,5)$. Geometrically, translating right 4 and down 2 increases $x$ by 4 and decreases $y$ by 2.