Question

apply the following transformations to the shape: first reflect over the y-axis, then translate 4 units right and 5 units up. show your work here

Explanation:

Step1: Identify Original Vertices

Let's take the blue rectangle (inner shape) vertices. From the graph, the blue vertices are at \((-3, 3)\), \((3, 3)\), \((3, -3)\), \((-3, -3)\)? Wait, no, looking at the grid, the blue points: left blue point is \((-3, 3)\)? Wait, no, the x - axis: the left blue dot is at \(x=-3\)? Wait, the grid lines: each square is 1 unit. Let's check the coordinates. The blue rectangle has vertices: let's see, the left - most blue point is at \((-3, 3)\)? Wait, no, the x - coordinate for the left blue dot: from the origin (0,0), moving left 3 units? Wait, the vertical line at \(x = - 3\)? Wait, no, the blue dots: the left blue dot is at \((-3, 3)\)? Wait, no, looking at the graph, the blue rectangle has vertices: \((-3, 3)\), \((3, 3)\), \((3, -3)\), \((-3, -3)\)? Wait, no, the y - axis: the top blue dot is at \(y = 3\), bottom at \(y=-3\), left at \(x=-3\), right at \(x = 3\)? Wait, no, the blue rectangle: left vertex \((-3, 3)\), right vertex \((3, 3)\), right - bottom \((3, -3)\), left - bottom \((-3, -3)\)? Wait, maybe I misread. Wait, the blue lines: the vertical lines are at \(x=-3\) and \(x = 3\)? No, the blue dots: the left blue dot is at \(x=-3\)? Wait, the grid: the x - axis has marks at - 14, - 12, - 10, - 8, - 6, - 4, - 2, 0, 2, 4, 6, etc. Wait, the blue dot on the left: it's at \(x=-3\)? Wait, no, the x - coordinate of the left blue dot: between - 4 and - 2, so \(x=-3\), y - coordinate: between 2 and 4, so \(y = 3\). Similarly, right blue dot: \(x = 3\), \(y = 3\), bottom right: \(x = 3\), \(y=-3\), bottom left: \(x=-3\), \(y=-3\). So original vertices (blue rectangle) are \(A(-3, 3)\), \(B(3, 3)\), \(C(3, -3)\), \(D(-3, -3)\). Wait, no, maybe the blue rectangle is smaller. Wait, the blue dots: the top blue line is at \(y = 3\)? Wait, the y - axis: the numbers are 2, 4, 6, etc. Wait, the blue dot's y - coordinate: the top blue dot is at \(y = 3\)? Wait, the grid: each square is 1 unit. Let's re - examine: the blue rectangle has vertices: let's take the top - left blue dot: \(x=-3\), \(y = 3\); top - right: \(x = 3\), \(y = 3\); bottom - right: \(x = 3\), \(y=-3\); bottom - left: \(x=-3\), \(y=-3\).

Step2: Reflect Over y - axis

The rule for reflection over the \(y\) - axis is \((x,y)\to(-x,y)\). So for each vertex:
- For \(A(-3, 3)\): reflect to \(A_1(3, 3)\)
- For \(B(3, 3)\): reflect to \(B_1(-3, 3)\)
- For \(C(3, -3)\): reflect to \(C_1(-3, -3)\)
- For \(D(-3, -3)\): reflect to \(D_1(3, -3)\)

Wait, that can't be right. Wait, maybe the original vertices are different. Wait, maybe the blue rectangle has vertices: let's look at the graph again. The blue dots: the left blue dot is at \(x=-3\)? Wait, no, the x - coordinate of the left blue dot: it's at \(x=-3\)? Wait, the vertical line at \(x=-3\) is between \(x=-4\) and \(x=-2\). Wait, maybe the original vertices (blue rectangle) are \(A(-3, 3)\), \(B(1, 3)\), \(C(1, -3)\), \(D(-3, -3)\)? No, that doesn't make sense. Wait, maybe I made a mistake. Let's take the blue rectangle: the top side is from \(x=-3\) to \(x = 1\) at \(y = 3\)? No, the blue line is horizontal, so same y - coordinate. Wait, the blue dots: the left blue dot is at \((-3, 3)\), right blue dot at \((1, 3)\)? No, the distance between the blue dots: from \(x=-3\) to \(x = 1\) is 4 units? No, the problem says "the shape" – maybe the black rectangle? Wait, the black rectangle has vertices: top - left \((-1, 6)\), top - right \((3, 6)\), bottom - right \((3, -6)\), bottom - left \((-1, -6)\). Wait, maybe I misidentified the shape. Let's check the black rectangle: top - left at \((-1, 6)\), top - right at \((3, 6)\), bottom - right at \((3, -6)\), bottom - left at \((-1, -6)\). And the blue rectangle: top - left \((-3, 3)\), top - right \((1, 3)\), bottom - right \((1, -3)\), bottom - left \((-3, -3)\). Wait, maybe the problem is about the blue rectangle. Let's proceed with the blue rectangle.

Original vertices of blue rectangle (let's correct): Let's look at the x - coordinates: the left blue dot is at \(x=-3\), right blue dot at \(x = 1\)? No, the horizontal line between the blue dots: from \(x=-3\) to \(x = 1\)? No, the grid lines: the blue dots are at \(x=-3\) and \(x = 1\)? Wait, no, the x - axis marks: - 4, - 2, 0, 2, 4. So the left blue dot is at \(x=-3\) (between - 4 and - 2), right blue dot at \(x = 1\) (between 0 and 2)? No, this is confusing. Alternatively, let's take a general point \((x,y)\) and apply the transformations.

Reflection over \(y\) - axis: \((x,y)\to(-x,y)\)

Translation: \((x,y)\to(x + 4,y + 5)\)

Let's take a vertex of the blue rectangle. Let's say the top - left vertex of the blue rectangle: from the graph, it's at \((-3, 3)\) (x=-3, y = 3).

Step3: Reflect the Vertex Over y - axis

For \((-3, 3)\), reflection over \(y\) - axis: \((-(-3),3)=(3, 3)\)

Step4: Translate the Reflected Vertex

Translate 4 units right (\(x+4\)) and 5 units up (\(y + 5\)): \((3 + 4,3+5)=(7, 8)\)

Take another vertex: top - right of blue rectangle: \((1, 3)\) (wait, no, maybe \(x = 3\)? Wait, the right blue dot is at \(x = 3\), \(y = 3\)). So original vertex \((3, 3)\).

Reflection over \(y\) - axis: \((-3, 3)\)

Translation: \((-3 + 4,3 + 5)=(1, 8)\)

Bottom - right vertex of blue rectangle: \((3, -3)\)

Reflection over \(y\) - axis: \((-3, -3)\)

Translation: \((-3+4,-3 + 5)=(1, 2)\)

Bottom - left vertex of blue rectangle: \((-3, -3)\)

Reflection over \(y\) - axis: \((3, -3)\)

Translation: \((3 + 4,-3+5)=(7, 2)\)

Now, let's plot these points: \((7, 8)\), \((1, 8)\), \((1, 2)\), \((7, 2)\). Connecting these points will give the transformed shape.

Answer:

To solve, we:

1. Reflect over \(y\) - axis: For a point \((x,y)\), use \((x,y)\to(-x,y)\).
2. Translate: For the reflected point \((-x,y)\), use \((-x + 4,y + 5)\) (4 right, 5 up).

Applying to vertices of the blue rectangle (e.g., \((-3,3)\), \((3,3)\), \((3,-3)\), \((-3,-3)\)):

• \((-3,3)\to(3,3)\to(7,8)\)
• \((3,3)\to(-3,3)\to(1,8)\)
• \((3,-3)\to(-3,-3)\to(1,2)\)
• \((-3,-3)\to(3,-3)\to(7,2)\)

Plot these points \((7,8)\), \((1,8)\), \((1,2)\), \((7,2)\) and connect them to get the transformed shape. (The final shape is a rectangle with vertices at \((1,8)\), \((7,8)\), \((7,2)\), \((1,2)\))