Math / Geometry

QuestionGraph $\overline{X Y}$ with endpoints $X(5,-2)$ and $Y(3,-3)$ and its image after a reflection in the $x$ -axis and then a rotation of $270^{\circ}$ counterclockwise about the origin.

Studdy Solution

STEP 1

What is this asking? We're going to reflect a line segment across the x-axis, then rotate it 270 degrees counterclockwise around the origin, and finally, we'll graph both the original and transformed line segments. Watch out! Don't mix up the order of the transformations!
We reflect then rotate.
Also, remember positive rotations are counterclockwise, and negative rotations are clockwise.

STEP 2

1. Reflect across the x-axis
2. Rotate around the origin
3. Graph the line segments

STEP 3

When we reflect a point $(x, y)$ across the x-axis, the x-coordinate stays the same, but the y-coordinate changes its sign.
So, reflecting $X(5, -2)$ across the x-axis gives us $X'(5, 2)$ .
We essentially multiplied the y-coordinate by $-1$ .

STEP 4

Similarly, reflecting $Y(3, -3)$ across the x-axis gives us $Y'(3, 3)$ .
Again, the y-coordinate was multiplied by $-1$ .

STEP 5

Remember, to rotate a point $(x, y)$ counterclockwise by 270 degrees around the origin, we use the formula $(x, y) \to (y, -x)$ .
This means our new x-coordinate is the old y-coordinate, and our new y-coordinate is the negative of the old x-coordinate.

STEP 6

Let's apply this to $X'(5, 2)$ .
Using our formula, we get $X''(2, -5)$ .
We swapped the coordinates and changed the sign of the original x-coordinate.

STEP 7

Now, let's rotate $Y'(3, 3)$ .
Using the same formula, we get $Y''(3, -3)$ .
We swapped the coordinates and changed the sign of the original x-coordinate.

STEP 8

First, plot the original points $X(5, -2)$ and $Y(3, -3)$ and connect them to form the line segment $\overline{XY}$ .

STEP 9

Next, plot the transformed points $X''(2, -5)$ and $Y''(3, -3)$ and connect them to form the transformed line segment $\overline{X''Y''}$ .

STEP 10

The original line segment $\overline{XY}$ has endpoints $X(5, -2)$ and $Y(3, -3)$ .
After reflecting across the x-axis and rotating 270 degrees counterclockwise around the origin, the transformed line segment $\overline{X''Y''}$ has endpoints $X''(2, -5)$ and $Y''(3, -3)$ .
Graph both line segments on the coordinate plane!

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QuestionGraph $\overline{X Y}$ with endpoints $X(5,-2)$ and $Y(3,-3)$ and its image after a reflection in the $x$ -axis and then a rotation of $270^{\circ}$ counterclockwise about the origin.

Studdy Solution

STEP 1

STEP 2

1. Reflect across the x-axis
2. Rotate around the origin
3. Graph the line segments

STEP 3

When we reflect a point $(x, y)$ across the x-axis, the x-coordinate stays the same, but the y-coordinate changes its sign.
So, reflecting $X(5, -2)$ across the x-axis gives us $X'(5, 2)$ .
We essentially multiplied the y-coordinate by $-1$ .

STEP 4

Similarly, reflecting $Y(3, -3)$ across the x-axis gives us $Y'(3, 3)$ .
Again, the y-coordinate was multiplied by $-1$ .

STEP 5

Remember, to rotate a point $(x, y)$ counterclockwise by 270 degrees around the origin, we use the formula $(x, y) \to (y, -x)$ .
This means our new x-coordinate is the old y-coordinate, and our new y-coordinate is the negative of the old x-coordinate.

STEP 6

Let's apply this to $X'(5, 2)$ .
Using our formula, we get $X''(2, -5)$ .
We swapped the coordinates and changed the sign of the original x-coordinate.

STEP 7

Now, let's rotate $Y'(3, 3)$ .
Using the same formula, we get $Y''(3, -3)$ .
We swapped the coordinates and changed the sign of the original x-coordinate.

STEP 8

First, plot the original points $X(5, -2)$ and $Y(3, -3)$ and connect them to form the line segment $\overline{XY}$ .

STEP 9

Next, plot the transformed points $X''(2, -5)$ and $Y''(3, -3)$ and connect them to form the transformed line segment $\overline{X''Y''}$ .

STEP 10

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QuestionGraph XY‾\overline{X Y}XY with endpoints X(5,−2)X(5,-2)X(5,−2) and Y(3,−3)Y(3,-3)Y(3,−3) and its image after a reflection in the xxx-axis and then a rotation of 270∘270^{\circ}270∘ counterclockwise about the origin.

Studdy Solution

STEP 1

STEP 2

1. Reflect across the x-axis2. Rotate around the origin3. Graph the line segments

STEP 3

When we reflect a point (x,y)(x, y)(x,y) across the x-axis, the x-coordinate stays the same, but the y-coordinate changes its sign.So, reflecting X(5,−2)X(5, -2)X(5,−2) across the x-axis gives us X′(5,2)X'(5, 2)X′(5,2).We essentially multiplied the y-coordinate by −1-1−1.

STEP 4

Similarly, reflecting Y(3,−3)Y(3, -3)Y(3,−3) across the x-axis gives us Y′(3,3)Y'(3, 3)Y′(3,3).Again, the y-coordinate was multiplied by −1-1−1.

STEP 5

Remember, to rotate a point (x,y)(x, y)(x,y) counterclockwise by 270 degrees around the origin, we use the formula (x,y)→(y,−x)(x, y) \to (y, -x)(x,y)→(y,−x).This means our new x-coordinate is the old y-coordinate, and our new y-coordinate is the negative of the old x-coordinate.

STEP 6

Let's apply this to X′(5,2)X'(5, 2)X′(5,2).Using our formula, we get X′′(2,−5)X''(2, -5)X′′(2,−5).We swapped the coordinates and changed the sign of the original x-coordinate.

STEP 7

Now, let's rotate Y′(3,3)Y'(3, 3)Y′(3,3).Using the same formula, we get Y′′(3,−3)Y''(3, -3)Y′′(3,−3).We swapped the coordinates and changed the sign of the original x-coordinate.

STEP 8

First, plot the original points X(5,−2)X(5, -2)X(5,−2) and Y(3,−3)Y(3, -3)Y(3,−3) and connect them to form the line segment XY‾\overline{XY}XY.

STEP 9

Next, plot the transformed points X′′(2,−5)X''(2, -5)X′′(2,−5) and Y′′(3,−3)Y''(3, -3)Y′′(3,−3) and connect them to form the transformed line segment X′′Y′′‾\overline{X''Y''}X′′Y′′.

STEP 10

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QuestionGraph $\overline{X Y}$ with endpoints $X(5,-2)$ and $Y(3,-3)$ and its image after a reflection in the $x$ -axis and then a rotation of $270^{\circ}$ counterclockwise about the origin.

1. Reflect across the x-axis
2. Rotate around the origin
3. Graph the line segments

When we reflect a point $(x, y)$ across the x-axis, the x-coordinate stays the same, but the y-coordinate changes its sign.
So, reflecting $X(5, -2)$ across the x-axis gives us $X'(5, 2)$ .
We essentially multiplied the y-coordinate by $-1$ .

Similarly, reflecting $Y(3, -3)$ across the x-axis gives us $Y'(3, 3)$ .
Again, the y-coordinate was multiplied by $-1$ .

Remember, to rotate a point $(x, y)$ counterclockwise by 270 degrees around the origin, we use the formula $(x, y) \to (y, -x)$ .
This means our new x-coordinate is the old y-coordinate, and our new y-coordinate is the negative of the old x-coordinate.

Let's apply this to $X'(5, 2)$ .
Using our formula, we get $X''(2, -5)$ .
We swapped the coordinates and changed the sign of the original x-coordinate.

Now, let's rotate $Y'(3, 3)$ .
Using the same formula, we get $Y''(3, -3)$ .
We swapped the coordinates and changed the sign of the original x-coordinate.

First, plot the original points $X(5, -2)$ and $Y(3, -3)$ and connect them to form the line segment $\overline{XY}$ .

Next, plot the transformed points $X''(2, -5)$ and $Y''(3, -3)$ and connect them to form the transformed line segment $\overline{X''Y''}$ .