To determine the correct transformation described by the equation [tex]\( y = x + 9.5 \)[/tex], we need to understand how the addition of a constant term to a function affects its graph. Here's the step-by-step explanation:
1. Start with the basic function: The original function given is [tex]\( y = x \)[/tex]. This is a straight line with a slope of 1 that passes through the origin (0,0).
2. Identify the transformation: The modified function is [tex]\( y = x + 9.5 \)[/tex]. This modification involves adding a constant, 9.5, to the original function.
3. Understand the effect of adding a constant:
- When you add a constant [tex]\( c \)[/tex] to a function [tex]\( f(x) \)[/tex], such as transforming [tex]\( y = f(x) \)[/tex] to [tex]\( y = f(x) + c \)[/tex], it results in a vertical shift.
- Specifically, adding a positive constant [tex]\( c \)[/tex] shifts the graph of the function up by [tex]\( c \)[/tex] units.
4. Apply this to our function:
- For [tex]\( y = x + 9.5 \)[/tex], the constant added is 9.5.
- Therefore, the graph of [tex]\( y = x \)[/tex] is shifted vertically upward by 9.5 units.
5. Verify the options:
- Option A states that the graph is translated 9.5 units up, which matches our explanation.
- Option B states that the graph is translated 9.5 units down, which is incorrect.
- Option C implies the slope is increased, but adding a constant does not affect the slope.
- Option D suggests the graph is translated horizontally to the right, which adding a constant does not support.
Given this careful analysis, the correct statement describing the graph of [tex]\( y = x + 9.5 \)[/tex] is:
A. It is the graph of [tex]\( y = x \)[/tex] translated 9.5 units up.
