Question

1. in the figure below, triangle efg is reflected across the horizontal line ( y = b ), where ( b ) is an integer. the coordinates of the image of point f are ( (-1, -5) ). what must be the value of ( b )?
2. figure irt and it’s image are graphed on the coordinate plane below. write the translation that occurred in words, and arrow notation.
3. michael’s teacher plotted figure a( (2, -4) ), b( (3, -1) ), c( (6, -1) ) and d( (5, -4) ). he then plotted a( (-3, -1) ). michael’s teacher then asked him to find the coordinate of b( ), based on the translation that determined a( ). michael determined that the coordinate of b( ) would be ( (7, -7) ). is michael correct? if so, explain why. if not, explain michael’s mistake, and give the correct coordinate for b( ).
4. which picture shows a ( 90^circ ) counterclockwise rotation of the following house:

a)

b)

c)

d)

1. write the following translation from arrow notation, to words

a) ( (x, y) \to (x + 4, y - 3) )
b) ( (x, y) \to (x - 6, y + 2) )
c) ( (x, y) \to (x - 1, y) )

Explanation:

Question 13

Step1: Identify original F coordinates

From the grid, original $F$ is $(-1, 3)$.

Step2: Use reflection midpoint formula

For reflection over $y=b$, the midpoint of $y$-coordinates equals $b$:
$$b = \frac{y_{\text{original}} + y_{\text{image}}}{2}$$

Step3: Substitute values and solve

$$b = \frac{3 + (-5)}{2} = \frac{-2}{2} = -1$$

1. Pick a point (e.g., $I(-2,7)$ and $I'(3,4)$). Calculate the change: $3 - (-2) = 5$ (right 5), $4 - 7 = -3$ (down 3).
2. Translate this to words and arrow notation.

Step1: Find translation rule

From $A(2,-4)$ to $A'(-3,-1)$:
$\Delta x = -3 - 2 = -5$, $\Delta y = -1 - (-4) = 3$
Rule: $(x,y) \to (x-5, y+3)$

Step2: Apply rule to B(3,-1)

$x' = 3 - 5 = -2$, $y' = -1 + 3 = 2$

Step3: Analyze Michael's error

Michael used $(x+5, y-3)$ (opposite of the correct translation).

Answer:

$-1$

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Question 14