Question

graph the image of $\triangle jkl$ after a rotation 90° counterclockwise around the origin and a translation 5 units down.

graph the image of $\triangle abc$ after a reflection over the x - axis and a translation 4 units right

Explanation:

First Sub - Question (Graphing $\triangle JKL$ after transformation)

Step 1: Identify coordinates of $\triangle JKL$

From the graph, we can see the coordinates: Let's assume \(J=(5,4)\), \(K=(1,5)\), \(L=(1,1)\) (we determine these by looking at the grid, where the x - coordinate is the horizontal position and y - coordinate is the vertical position).

Step 2: Perform 90° counter - clockwise rotation about the origin

The rule for a 90° counter - clockwise rotation about the origin \((x,y)\to(-y,x)\)

• For \(J=(5,4)\): After rotation, \(J'=(- 4,5)\)
• For \(K=(1,5)\): After rotation, \(K'=(-5,1)\)
• For \(L=(1,1)\): After rotation, \(L'=(-1,1)\)

Step 3: Perform translation 5 units down

The rule for a translation 5 units down is \((x,y)\to(x,y - 5)\)

• For \(J'=(-4,5)\): After translation, \(J''=(-4,5 - 5)=(-4,0)\)
• For \(K'=(-5,1)\): After translation, \(K''=(-5,1 - 5)=(-5,-4)\)
• For \(L'=(-1,1)\): After translation, \(L''=(-1,1 - 5)=(-1,-4)\)

Step 4: Graph the new triangle

Plot the points \(J''(-4,0)\), \(K''(-5,-4)\), \(L''(-1,-4)\) and connect them to get the image of \(\triangle JKL\) after the transformation.

Second Sub - Question (Graphing $\triangle ABC$ after transformation)

Step 1: Identify coordinates of $\triangle ABC$

From the graph, we can see the coordinates: Let's assume \(A=(3,-5)\), \(B=(1,-2)\), \(C=(1,-5)\) (by looking at the grid, x - coordinate is horizontal and y - coordinate is vertical).

Step 2: Perform reflection over the x - axis

The rule for reflection over the x - axis is \((x,y)\to(x,-y)\)

• For \(A=(3,-5)\): After reflection, \(A'=(3,5)\)
• For \(B=(1,-2)\): After reflection, \(B'=(1,2)\)
• For \(C=(1,-5)\): After reflection, \(C'=(1,5)\)

Step 3: Perform translation 4 units right

The rule for a translation 4 units right is \((x,y)\to(x + 4,y)\)

• For \(A'=(3,5)\): After translation, \(A''=(3 + 4,5)=(7,5)\)
• For \(B'=(1,2)\): After translation, \(B''=(1+4,2)=(5,2)\)
• For \(C'=(1,5)\): After translation, \(C''=(1 + 4,5)=(5,5)\)

Step 4: Graph the new triangle

Plot the points \(A''(7,5)\), \(B''(5,2)\), \(C''(5,5)\) and connect them to get the image of \(\triangle ABC\) after the transformation.

Final Answer (Graphical Description)

• For \(\triangle JKL\): The image after 90° counter - clockwise rotation about the origin and 5 units down translation has vertices at \((-4,0)\), \((-5,-4)\), \((-1,-4)\).
• For \(\triangle ABC\): The image after reflection over the x - axis and 4 units right translation has vertices at \((7,5)\), \((5,2)\), \((5,5)\).

(Note: Since the problem asks to graph, the key is to find the new coordinates as above and plot them. The actual graphing is done on the coordinate plane by marking the new points and connecting them.)

Answer:

First Sub - Question (Graphing $\triangle JKL$ after transformation)

Step 1: Identify coordinates of $\triangle JKL$

From the graph, we can see the coordinates: Let's assume \(J=(5,4)\), \(K=(1,5)\), \(L=(1,1)\) (we determine these by looking at the grid, where the x - coordinate is the horizontal position and y - coordinate is the vertical position).

Step 2: Perform 90° counter - clockwise rotation about the origin

The rule for a 90° counter - clockwise rotation about the origin \((x,y)\to(-y,x)\)

• For \(J=(5,4)\): After rotation, \(J'=(- 4,5)\)
• For \(K=(1,5)\): After rotation, \(K'=(-5,1)\)
• For \(L=(1,1)\): After rotation, \(L'=(-1,1)\)

Step 3: Perform translation 5 units down

The rule for a translation 5 units down is \((x,y)\to(x,y - 5)\)

• For \(J'=(-4,5)\): After translation, \(J''=(-4,5 - 5)=(-4,0)\)
• For \(K'=(-5,1)\): After translation, \(K''=(-5,1 - 5)=(-5,-4)\)
• For \(L'=(-1,1)\): After translation, \(L''=(-1,1 - 5)=(-1,-4)\)

Step 4: Graph the new triangle

Plot the points \(J''(-4,0)\), \(K''(-5,-4)\), \(L''(-1,-4)\) and connect them to get the image of \(\triangle JKL\) after the transformation.

Second Sub - Question (Graphing $\triangle ABC$ after transformation)

Step 1: Identify coordinates of $\triangle ABC$

From the graph, we can see the coordinates: Let's assume \(A=(3,-5)\), \(B=(1,-2)\), \(C=(1,-5)\) (by looking at the grid, x - coordinate is horizontal and y - coordinate is vertical).

Step 2: Perform reflection over the x - axis

The rule for reflection over the x - axis is \((x,y)\to(x,-y)\)

• For \(A=(3,-5)\): After reflection, \(A'=(3,5)\)
• For \(B=(1,-2)\): After reflection, \(B'=(1,2)\)
• For \(C=(1,-5)\): After reflection, \(C'=(1,5)\)

Step 3: Perform translation 4 units right

The rule for a translation 4 units right is \((x,y)\to(x + 4,y)\)

• For \(A'=(3,5)\): After translation, \(A''=(3 + 4,5)=(7,5)\)
• For \(B'=(1,2)\): After translation, \(B''=(1+4,2)=(5,2)\)
• For \(C'=(1,5)\): After translation, \(C''=(1 + 4,5)=(5,5)\)

Step 4: Graph the new triangle

Plot the points \(A''(7,5)\), \(B''(5,2)\), \(C''(5,5)\) and connect them to get the image of \(\triangle ABC\) after the transformation.

Final Answer (Graphical Description)

• For \(\triangle JKL\): The image after 90° counter - clockwise rotation about the origin and 5 units down translation has vertices at \((-4,0)\), \((-5,-4)\), \((-1,-4)\).
• For \(\triangle ABC\): The image after reflection over the x - axis and 4 units right translation has vertices at \((7,5)\), \((5,2)\), \((5,5)\).

(Note: Since the problem asks to graph, the key is to find the new coordinates as above and plot them. The actual graphing is done on the coordinate plane by marking the new points and connecting them.)