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Each statement describes a transformation of the graph of [tex]\( y = x \)[/tex]. Which statement correctly describes the graph of [tex]\( y = x + 9.5 \)[/tex]?

A. It is the graph of [tex]\( y = x \)[/tex] translated 9.5 units to the right.
B. It is the graph of [tex]\( y = x \)[/tex] where the slope is increased by 9.5.
C. It is the graph of [tex]\( y = x \)[/tex] translated 9.5 units down.
D. It is the graph of [tex]\( y = x \)[/tex] translated 9.5 units up.

Sagot :

GinnyAnsweridn
To determine which transformation correctly describes the graph of [tex]\( y = x + 9.5 \)[/tex], let's analyze the equation step by step.

1. Understanding the equation [tex]\( y = x + 9.5 \)[/tex]:
- The base equation is [tex]\( y = x \)[/tex], which is a straight line passing through the origin (0,0) with a slope of 1.
- The term [tex]\( +9.5 \)[/tex] is a constant added to the [tex]\( y \)[/tex]-value of every point on the line [tex]\( y = x \)[/tex].

2. Impact of adding a constant to [tex]\( y \)[/tex]:
- Adding a constant value to the [tex]\( y \)[/tex] in the equation of a line results in a vertical translation.
- Specifically, adding a positive constant shifts the graph up by that constant value, and adding a negative constant shifts it down.

3. Application to the given equation:
- In the given equation [tex]\( y = x + 9.5 \)[/tex], the positive constant 9.5 added to [tex]\( y \)[/tex] means we translate the graph of [tex]\( y = x \)[/tex] 9.5 units up vertically.
- This doesn't change the slope of the line, only its position along the [tex]\( y \)[/tex]-axis.

4. Comparing options:
- Option A: Translation 9.5 units to the right implies a horizontal shift, not correct in this context.
- Option B: Slope change implies altering the steepness/incline of the graph, which is not the case here.
- Option C: Translation 9.5 units down would imply [tex]\( y = x - 9.5 \)[/tex], which is not our equation.
- Option D: Translation 9.5 units up matches perfectly with the transformation described by [tex]\( y = x + 9.5 \)[/tex].

Therefore, the correct description of the graph of [tex]\( y = x + 9.5 \)[/tex] is:

D. It is the graph of [tex]\( y = x \)[/tex] translated 9.5 units up.
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