To determine how the graph of the function [tex]\( h(x) = g(x) + 1 \)[/tex] relates to the graph of the function [tex]\( g(x) = x^2 \)[/tex], we need to analyze the transformation step by step.
1. Starting Point:
[tex]\( g(x) = x^2 \)[/tex]
This is a standard parabola opening upwards with its vertex at the origin (0, 0).
2. Defining the Transformation:
The function [tex]\( h(x) \)[/tex] is defined as:
[tex]\[
h(x) = g(x) + 1
\][/tex]
Substituting [tex]\( g(x) \)[/tex] into the equation for [tex]\( h(x) \)[/tex]:
[tex]\[
h(x) = x^2 + 1
\][/tex]
This shows that [tex]\( h(x) \)[/tex] takes the value of [tex]\( g(x) \)[/tex] (which is [tex]\( x^2 \)[/tex]) and adds 1 to it.
3. Understanding the Transformation:
In general, adding a constant [tex]\( k \)[/tex] to a function [tex]\( g(x) \)[/tex] (i.e., [tex]\( h(x) = g(x) + k \)[/tex]) results in a vertical shift of the graph of [tex]\( g(x) \)[/tex] by [tex]\( k \)[/tex] units. If [tex]\( k \)[/tex] is positive, the shift is upwards; if [tex]\( k \)[/tex] is negative, the shift is downwards.
4. Applying to Our Case:
Here, since [tex]\( k = 1 \)[/tex], the graph of [tex]\( g(x) = x^2 \)[/tex] will be shifted vertically upwards by 1 unit to obtain [tex]\( h(x) = x^2 + 1 \)[/tex].
5. Verifying with Points:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = 0^2 = 0 \quad \text{and} \quad h(0) = 0^2 + 1 = 1
\][/tex]
The point (0, 0) on the graph of [tex]\( g(x) \)[/tex] is shifted to (0, 1) on the graph of [tex]\( h(x) \)[/tex].
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = 1^2 = 1 \quad \text{and} \quad h(1) = 1^2 + 1 = 2
\][/tex]
The point (1, 1) on the graph of [tex]\( g(x) \)[/tex] is shifted to (1, 2) on the graph of [tex]\( h(x) \)[/tex].
Similar shifts can be observed for other points as well.
6. Conclusion:
Based on the analysis, we can conclude the correct answer:
[tex]\[
\text{A. The graph of } h \text{ is the graph of } g \text{ vertically shifted up 1 unit.}
\][/tex]
Therefore, the correct statement is:
A. The graph of [tex]\( h \)[/tex] is the graph of [tex]\( g \)[/tex] vertically shifted up 1 unit.
