Question

use the graph below to add points to the table below.

x
y

a) write the rule in words.

b) explain how to use the table to predict the value of y when x is -8.

6-97. use these following directions to create a mystery letter. plot these points and connect them in order to create a rectangle: (2,1),(2,4),(3,4),(3,1).

be sure to connect the last point to the first point. then follow the directions in parts (a) through (c) below.

a) rotate the rectangle 90° clockwise (↻) about the point (2,1) and draw the rotated rectangle.

b) reflect the new rectangle over the line y = 2.5 and draw the reflected rectangle.

c) name the letter of the alphabet that your graph resembles.

Explanation:

Part 6-97:

a) Rotating the Rectangle 90° Clockwise about (2,1)

To rotate a point \((x,y)\) 90° clockwise about a center \((h,k)\), we use the transformation:
New \(x = h + (y - k)\)
New \(y = k - (x - h)\)

1. For point (2,1) (center):

Since it's the center, it remains \((2,1)\).

1. For point (2,4):

\(h = 2\), \(k = 1\), \(x = 2\), \(y = 4\)
New \(x = 2 + (4 - 1) = 5\)
New \(y = 1 - (2 - 2) = 1\)
Rotated point: \((5,1)\)

1. For point (3,4):

\(x = 3\), \(y = 4\)
New \(x = 2 + (4 - 1) = 5\)
New \(y = 1 - (3 - 2) = 0\)
Rotated point: \((5,0)\)

1. For point (3,1):

\(x = 3\), \(y = 1\)
New \(x = 2 + (1 - 1) = 2\)
New \(y = 1 - (3 - 2) = 0\)
Rotated point: \((2,0)\)

The rotated rectangle has vertices: \((2,1)\), \((5,1)\), \((5,0)\), \((2,0)\).

b) Reflecting Over \(y = 2.5\)

To reflect a point \((x,y)\) over \(y = c\), the new \(y\)-coordinate is \(2c - y\), and \(x\) remains. Here, \(c = 2.5\).

1. For (2,1):

New \(y = 2(2.5) - 1 = 4\)
Reflected point: \((2,4)\)

1. For (5,1):

New \(y = 2(2.5) - 1 = 4\)
Reflected point: \((5,4)\)

1. For (5,0):

New \(y = 2(2.5) - 0 = 5\)
Reflected point: \((5,5)\)

1. For (2,0):

New \(y = 2(2.5) - 0 = 5\)
Reflected point: \((2,5)\)

The reflected rectangle has vertices: \((2,4)\), \((5,4)\), \((5,5)\), \((2,5)\).

c) Naming the Letter

After plotting the original rectangle, rotating, reflecting, and connecting, the combined shape (original + transformed) resembles the letter "H" (or may form a vertical/horizontal bar structure typical of "H").

For the Graph and Table (Top Section):

Step 1: Identify Points on the Line

From the graph, we can pick \(x\)-values and find corresponding \(y\)-values. For example:

• When \(x = 0\), \(y = 5\) (intercept)
• When \(x = 3\), \(y = 2\)
• When \(x = 4\), \(y = 1\)
• When \(x = 5\), \(y = 0\) (intercept)
• When \(x = -2\), \(y = 8\) (since slope \(m = \frac{2 - 5}{3 - 0} = -1\), so \(y = -x + 5\))

a) Rule in Words

The \(y\)-value is equal to 5 minus the \(x\)-value (or \(y = -x + 5\)).

b) Predict \(y\) when \(x = -8\)

Using the rule \(y = -x + 5\):
Substitute \(x = -8\):
\(y = -(-8) + 5 = 8 + 5 = 13\).

Final Answers:

• 6-97c: The graph resembles the letter \(\boldsymbol{H}\) (or similar, depending on precise plotting).
• Table Points (example):

\(x\): \(-2\), \(0\), \(3\), \(4\), \(5\)
\(y\): \(8\), \(5\), \(2\), \(1\), \(0\)

• a) Rule: \(y\) is 5 minus \(x\).
• b) \(y = 13\) when \(x = -8\).

Answer:

Part 6-97:

a) Rotating the Rectangle 90° Clockwise about (2,1)

To rotate a point \((x,y)\) 90° clockwise about a center \((h,k)\), we use the transformation:
New \(x = h + (y - k)\)
New \(y = k - (x - h)\)

1. For point (2,1) (center):

Since it's the center, it remains \((2,1)\).

1. For point (2,4):

\(h = 2\), \(k = 1\), \(x = 2\), \(y = 4\)
New \(x = 2 + (4 - 1) = 5\)
New \(y = 1 - (2 - 2) = 1\)
Rotated point: \((5,1)\)

1. For point (3,4):

\(x = 3\), \(y = 4\)
New \(x = 2 + (4 - 1) = 5\)
New \(y = 1 - (3 - 2) = 0\)
Rotated point: \((5,0)\)

1. For point (3,1):

\(x = 3\), \(y = 1\)
New \(x = 2 + (1 - 1) = 2\)
New \(y = 1 - (3 - 2) = 0\)
Rotated point: \((2,0)\)

The rotated rectangle has vertices: \((2,1)\), \((5,1)\), \((5,0)\), \((2,0)\).

b) Reflecting Over \(y = 2.5\)

To reflect a point \((x,y)\) over \(y = c\), the new \(y\)-coordinate is \(2c - y\), and \(x\) remains. Here, \(c = 2.5\).

1. For (2,1):

New \(y = 2(2.5) - 1 = 4\)
Reflected point: \((2,4)\)

1. For (5,1):

New \(y = 2(2.5) - 1 = 4\)
Reflected point: \((5,4)\)

1. For (5,0):

New \(y = 2(2.5) - 0 = 5\)
Reflected point: \((5,5)\)

1. For (2,0):

New \(y = 2(2.5) - 0 = 5\)
Reflected point: \((2,5)\)

The reflected rectangle has vertices: \((2,4)\), \((5,4)\), \((5,5)\), \((2,5)\).

c) Naming the Letter

After plotting the original rectangle, rotating, reflecting, and connecting, the combined shape (original + transformed) resembles the letter "H" (or may form a vertical/horizontal bar structure typical of "H").

For the Graph and Table (Top Section):

Step 1: Identify Points on the Line

From the graph, we can pick \(x\)-values and find corresponding \(y\)-values. For example:

• When \(x = 0\), \(y = 5\) (intercept)
• When \(x = 3\), \(y = 2\)
• When \(x = 4\), \(y = 1\)
• When \(x = 5\), \(y = 0\) (intercept)
• When \(x = -2\), \(y = 8\) (since slope \(m = \frac{2 - 5}{3 - 0} = -1\), so \(y = -x + 5\))

a) Rule in Words

The \(y\)-value is equal to 5 minus the \(x\)-value (or \(y = -x + 5\)).

b) Predict \(y\) when \(x = -8\)

Using the rule \(y = -x + 5\):
Substitute \(x = -8\):
\(y = -(-8) + 5 = 8 + 5 = 13\).

Final Answers:

• 6-97c: The graph resembles the letter \(\boldsymbol{H}\) (or similar, depending on precise plotting).
• Table Points (example):

\(x\): \(-2\), \(0\), \(3\), \(4\), \(5\)
\(y\): \(8\), \(5\), \(2\), \(1\), \(0\)

• a) Rule: \(y\) is 5 minus \(x\).
• b) \(y = 13\) when \(x = -8\).