To determine the effect of the number 1 on the graph of the function [tex]\( y - 3 = 2(x - 1)^2 \)[/tex] in comparison to the standard graph of [tex]\( v = x^2 \)[/tex], we need to analyze the transformation.
Given the standard form of a quadratic function:
[tex]\[ y = a(x - h)^2 + k \][/tex]
Here, [tex]\( (h, k) \)[/tex] represents the vertex of the parabola, and the graph of [tex]\( y = (x - h)^2 + k \)[/tex] is derived from [tex]\( y = x^2 \)[/tex] by shifting it horizontally by [tex]\( h \)[/tex] units and vertically by [tex]\( k \)[/tex] units.
Let's rewrite the given function in a form that is easier to compare with the standard form:
[tex]\[ y - 3 = 2(x - 1)^2 \][/tex]
First, we recognize that the expression can be seen as:
[tex]\[ y = 2(x - 1)^2 + 3 \][/tex]
From this expression, we can identify [tex]\( h \)[/tex] and [tex]\( k \)[/tex]:
- [tex]\( h = 1 \)[/tex]
- [tex]\( k = 3 \)[/tex]
The value of [tex]\( h \)[/tex] in the term [tex]\( (x - 1) \)[/tex] indicates a horizontal shift.
- If [tex]\( h \)[/tex] is positive, the graph shifts [tex]\( h \)[/tex] units to the right.
- If [tex]\( h \)[/tex] is negative, the graph shifts [tex]\( |h| \)[/tex] units to the left.
Since [tex]\( h = 1 \)[/tex], the graph of [tex]\( y = 2(x - 1)^2 + 3 \)[/tex] shifts 1 unit to the right in comparison to [tex]\( y = x^2 \)[/tex].
So, the effect of the number 1 on the graph is a shift of 1 unit to the right.
Thus, the correct answer is:
A. It shifts the graph 1 unit to the right.
