Question

graph the image of δghi (or similar triangle) after the following sequence of transformations: reflection across the x-axis, translation 8 units left and 17 units down (text with grid image of triangle)

Explanation:

To solve the problem of graphing the image of \(\triangle GHJ\) after the given transformations, we first need to identify the coordinates of the vertices of \(\triangle GHJ\). From the graph, let's assume the coordinates of \(G\), \(H\), and \(J\) are:

• \(G\): Let's say \(G = (x_1, y_1)\)
• \(H\): Let's say \(H = (x_2, y_2)\)
• \(J\): Let's say \(J = (x_3, y_3)\)

Looking at the graph, we can estimate the coordinates (assuming each grid square is 1 unit):

• \(J\) appears to be at \((8, -14)\)
• \(H\) appears to be at \((10, -11)\)
• \(G\) appears to be at \((12, -14)\)

Step 1: Reflect over the \(x\)-axis

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

• For \(J(8, -14)\): Reflecting over the \(x\)-axis gives \((8, 14)\)
• For \(H(10, -11)\): Reflecting over the \(x\)-axis gives \((10, 11)\)
• For \(G(12, -14)\): Reflecting over the \(x\)-axis gives \((12, 14)\)

Step 2: Translate 8 units left and 17 units down

The rule for translating a point \((x, y)\) \(a\) units left and \(b\) units down is \((x - a, y - b)\). Here, \(a = 8\) and \(b = 17\).

• For the reflected \(J(8, 14)\): Translating 8 units left and 17 units down gives \((8 - 8, 14 - 17) = (0, -3)\)
• For the reflected \(H(10, 11)\): Translating 8 units left and 17 units down gives \((10 - 8, 11 - 17) = (2, -6)\)
• For the reflected \(G(12, 14)\): Translating 8 units left and 17 units down gives \((12 - 8, 14 - 17) = (4, -3)\)

Step 3: Graph the new points

Plot the points \((0, -3)\), \((2, -6)\), and \((4, -3)\) and connect them to form the image of \(\triangle GHJ\) after the transformations.

Final Answer

The coordinates of the image of \(\triangle GHJ\) after the transformations are \(J'(0, -3)\), \(H'(2, -6)\), and \(G'(4, -3)\). The graph should show these points connected to form the transformed triangle.

Answer:

To solve the problem of graphing the image of \(\triangle GHJ\) after the given transformations, we first need to identify the coordinates of the vertices of \(\triangle GHJ\). From the graph, let's assume the coordinates of \(G\), \(H\), and \(J\) are:

• \(G\): Let's say \(G = (x_1, y_1)\)
• \(H\): Let's say \(H = (x_2, y_2)\)
• \(J\): Let's say \(J = (x_3, y_3)\)

Looking at the graph, we can estimate the coordinates (assuming each grid square is 1 unit):

• \(J\) appears to be at \((8, -14)\)
• \(H\) appears to be at \((10, -11)\)
• \(G\) appears to be at \((12, -14)\)

Step 1: Reflect over the \(x\)-axis

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

• For \(J(8, -14)\): Reflecting over the \(x\)-axis gives \((8, 14)\)
• For \(H(10, -11)\): Reflecting over the \(x\)-axis gives \((10, 11)\)
• For \(G(12, -14)\): Reflecting over the \(x\)-axis gives \((12, 14)\)

Step 2: Translate 8 units left and 17 units down

The rule for translating a point \((x, y)\) \(a\) units left and \(b\) units down is \((x - a, y - b)\). Here, \(a = 8\) and \(b = 17\).

• For the reflected \(J(8, 14)\): Translating 8 units left and 17 units down gives \((8 - 8, 14 - 17) = (0, -3)\)
• For the reflected \(H(10, 11)\): Translating 8 units left and 17 units down gives \((10 - 8, 11 - 17) = (2, -6)\)
• For the reflected \(G(12, 14)\): Translating 8 units left and 17 units down gives \((12 - 8, 14 - 17) = (4, -3)\)

Step 3: Graph the new points

Plot the points \((0, -3)\), \((2, -6)\), and \((4, -3)\) and connect them to form the image of \(\triangle GHJ\) after the transformations.

Final Answer

The coordinates of the image of \(\triangle GHJ\) after the transformations are \(J'(0, -3)\), \(H'(2, -6)\), and \(G'(4, -3)\). The graph should show these points connected to form the transformed triangle.