To determine which statement correctly describes the transformation of the graph of [tex]\( y = x \)[/tex] to [tex]\( y = x - 13 \)[/tex], let's go through the options one by one and understand what each transformation means for the graph:
1. Option A: It is the graph of [tex]\( y = x \)[/tex] translated 13 units up.
Translating the graph of [tex]\( y = x \)[/tex] up by 13 units would result in the equation [tex]\( y = x + 13 \)[/tex]. This adjustment increases every y-value by 13 units.
2. Option B: It is the graph of [tex]\( y = x \)[/tex] translated 13 units to the right.
Translating the graph of [tex]\( y = x \)[/tex] to the right by 13 units affects the x-values. This results in the equation [tex]\( y = (x - 13) \)[/tex]. However, this is not the equation we have.
3. Option C: It is the graph of [tex]\( y = x \)[/tex] translated 13 units to the left.
Translating the graph of [tex]\( y = x \)[/tex] to the left by 13 units would result in the equation [tex]\( y = (x + 13) \)[/tex]. This adjustment decreases every x-value by 13 units.
4. Option D: It is the graph of [tex]\( y = x \)[/tex] where the slope is decreased by 13.
The slope of the line [tex]\( y = x \)[/tex] is 1. Decreasing this slope by 13 units would give a new slope of [tex]\( 1 - 13 = -12 \)[/tex]. So, the equation would be something like [tex]\( y = -12x \)[/tex], which does not match [tex]\( y = x - 13 \)[/tex].
Given our goal is to match the equation [tex]\( y = x - 13 \)[/tex]:
- Translating the graph down by 13 units means every y-value is decreased by 13 units, which fits [tex]\( y = x - 13 \)[/tex].
The correct statement, therefore, is:
A. It is the graph of [tex]\( y = x \)[/tex] translated 13 units down.
