To determine the graph of the given function, [tex]\( g(x) = (x + 7)^2 \)[/tex], using transformations of the basic quadratic function [tex]\( f(x) = x^2 \)[/tex], follow these steps:
1. Identify the basic graph:
The basic function [tex]\( f(x) = x^2 \)[/tex] is a standard parabola that opens upwards with its vertex at the origin [tex]\((0, 0)\)[/tex].
2. Understand the transformation:
The function [tex]\( g(x) = (x + 7)^2 \)[/tex] can be viewed as a transformation of [tex]\( f(x) \)[/tex]. Specifically, it is a horizontal shift.
3. Determine the type of transformation:
- The expression inside the parentheses, [tex]\((x + 7)\)[/tex], indicates a horizontal shift.
- When a function [tex]\( f(x)\)[/tex] is modified to [tex]\( f(x + c) \)[/tex], this represents a shift to the left by [tex]\( c \)[/tex] units if [tex]\( c \)[/tex] is positive.
4. Apply the transformation:
- Since [tex]\( g(x) = (x + 7)^2 \)[/tex], we are shifting the graph of [tex]\( f(x) = x^2 \)[/tex] to the left by 7 units.
5. Describe the resulting graph:
- The graph of [tex]\( g(x) = (x + 7)^2 \)[/tex] is a parabola that opens upwards, similar in shape to the graph of [tex]\( f(x) = x^2 \)[/tex].
- The primary difference is that the vertex of the parabola [tex]\( g(x) = (x + 7)^2 \)[/tex] is shifted leftward by 7 units compared to [tex]\( f(x) = x^2 \)[/tex].
- Therefore, the vertex of [tex]\( g(x) = (x + 7)^2 \)[/tex] is at [tex]\((-7, 0)\)[/tex] instead of [tex]\((0, 0)\)[/tex].
In summary, the graph of [tex]\( g(x) = (x + 7)^2 \)[/tex] is a horizontal translation of the graph of [tex]\( f(x) = x^2 \)[/tex] to the left by 7 units. The shape remains the same, with the vertex of the parabola moving from [tex]\((0, 0)\)[/tex] to [tex]\((-7, 0)\)[/tex].
