To determine the graph of the given function [tex]\( g(x) = (x + 7)^2 \)[/tex] using transformations of the graph of the basic function [tex]\( f(x) = x^2 \)[/tex], follow these steps:
1. Identify the Basic Function:
The basic function given is [tex]\( f(x) = x^2 \)[/tex], which is a parabola opening upwards with its vertex at the origin, [tex]\((0, 0)\)[/tex].
2. Rewrite the Given Function in Terms of the Basic Function:
We want to express [tex]\( g(x) \)[/tex] in a form that clearly shows its relationship to [tex]\( f(x) \)[/tex]:
[tex]\[
g(x) = (x + 7)^2
\][/tex]
Notice that [tex]\( g(x) \)[/tex] looks like [tex]\( f(x) \)[/tex], but instead of [tex]\( x^2 \)[/tex], we have [tex]\((x + 7)^2\)[/tex].
3. Determine the Transformation:
To determine how the graph of [tex]\( f(x) \)[/tex] has been transformed to obtain [tex]\( g(x) \)[/tex], we need to analyze the expression [tex]\((x + 7)^2\)[/tex]:
- Inside the parentheses, we have [tex]\( x + 7 \)[/tex], which means that every [tex]\( x \)[/tex] value is increased by 7 before squaring. In graphical terms, this results in a horizontal transformation.
4. Identify the Type of Horizontal Transformation:
When we add a positive number (e.g., 7) inside the function's argument, it causes a horizontal shift to the left. This might seem counterintuitive because we are adding a positive number, but the shift occurs in the opposite direction (towards the left).
- The function [tex]\( (x + 7) \)[/tex] means that the graph of [tex]\( f(x) \)[/tex] is shifted 7 units to the left.
5. Summarize the Transformation:
The graph of the function [tex]\( g(x) = (x + 7)^2 \)[/tex] is obtained by taking the graph of the basic function [tex]\( f(x) = x^2 \)[/tex] and shifting it horizontally 7 units to the left.
6. Visual Representation:
- The vertex of the original function [tex]\( f(x) = x^2 \)[/tex] is at [tex]\((0, 0)\)[/tex].
- After the transformation (a shift to the left by 7 units), the vertex of [tex]\( g(x) = (x + 7)^2 \)[/tex] is at [tex]\((-7, 0)\)[/tex].
Therefore, the transformation required to obtain the graph of [tex]\( g(x) = (x + 7)^2 \)[/tex] from the graph of [tex]\( f(x) = x^2 \)[/tex] is a horizontal shift of the graph of [tex]\( f(x) \)[/tex] by 7 units to the left.
