To determine the effect on the graph of the function [tex]\(y = x^2 + 5x - 2\)[/tex] when [tex]\((x - 3)\)[/tex] is substituted for [tex]\(x\)[/tex], we need to understand how the transformation of the variable [tex]\(x\)[/tex] affects the graph.
1. Start with the original function:
[tex]\[
y = x^2 + 5x - 2
\][/tex]
2. Substitute [tex]\((x - 3)\)[/tex] in place of [tex]\(x\)[/tex]:
[tex]\[
y = (x - 3)^2 + 5(x - 3) - 2
\][/tex]
3. Expand the expression [tex]\((x - 3)^2\)[/tex]:
[tex]\[
(x - 3)^2 = x^2 - 6x + 9
\][/tex]
4. Expand the expression [tex]\(5(x - 3)\)[/tex]:
[tex]\[
5(x - 3) = 5x - 15
\][/tex]
5. Substitute these expanded expressions back into the modified function:
[tex]\[
y = (x^2 - 6x + 9) + (5x - 15) - 2
\][/tex]
6. Combine like terms:
[tex]\[
y = x^2 - 6x + 9 + 5x - 15 - 2
\][/tex]
[tex]\[
y = x^2 - x - 8
\][/tex]
Notice that the resulting function after substitution and simplification is [tex]\(y = x^2 - x - 8\)[/tex]. However, the important observation here is the horizontal shift that occurs when substituting [tex]\((x - 3)\)[/tex] for [tex]\(x\)[/tex].
Analysis:
- Substituting [tex]\((x - 3)\)[/tex] for [tex]\(x\)[/tex] results in a horizontal shift of the graph to the right by 3 units. This is a general property of function transformation where replacing [tex]\(x\)[/tex] with [tex]\((x - h)\)[/tex] shifts the graph of the function to the right by [tex]\(h\)[/tex] units.
Conclusion:
Accordingly, the correct answer is:
B. The graph would shift 3 units to the right.
