To compare the graphs of the functions [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = 7x^2 \)[/tex], let’s carefully examine each given statement.
1. Identifying the Functions:
- [tex]\( f(x) = x^2 \)[/tex]: This is a standard quadratic function that graphs as a parabola opening upwards.
- [tex]\( g(x) = 7x^2 \)[/tex]: This is another quadratic function, but with a coefficient of 7 multiplying the [tex]\( x^2 \)[/tex] term.
2. Comparing Transformations:
- Vertical Shifts:
- Shifting a graph vertically involves adding or subtracting a constant to the entire function. For example, [tex]\( f(x) + 7 = x^2 + 7 \)[/tex] would shift the graph upward by 7 units. However, [tex]\( g(x) = 7x^2 \)[/tex] does not involve adding or subtracting a constant; it instead involves multiplication.
- Vertical Stretches and Compressions:
- A vertical stretch or compression involves multiplying the function by a constant. If we multiply [tex]\( f(x) \)[/tex] by a factor greater than 1, the graph will stretch vertically. Specifically, if [tex]\( g(x) = 7x^2 \)[/tex], then this means that for every value of [tex]\( x \)[/tex], [tex]\( g(x) \)[/tex] is 7 times [tex]\( f(x) \)[/tex]. Therefore, the graph of [tex]\( g(x) \)[/tex] is vertically stretched by a factor of 7 compared to the graph of [tex]\( f(x) \)[/tex].
- Vertical compression would occur if we multiplied the function by a factor between 0 and 1. However, 7 is greater than 1, indicating a stretch, not a compression.
- Horizontal Stretches and Compressions:
- Horizontal stretches or compressions involve multiplying the [tex]\( x \)[/tex] inside the function by a constant. For example, [tex]\( f(kx) \)[/tex] would either stretch or compress the graph horizontally depending on the value of [tex]\( k \)[/tex]. Since [tex]\( g(x) = 7x^2 \)[/tex] does not modify the [tex]\( x \)[/tex] value inside the function, this change is not horizontal.
Given these comparisons, we can conclusively determine which statement correctly describes the transformation from [tex]\( f(x) = x^2 \)[/tex] to [tex]\( g(x) = 7x^2 \)[/tex]:
- Option A: Incorrect, shifting vertically implies adding a constant.
- Option B: Incorrect, [tex]\(g(x)\)[/tex] does not compress the graph, it stretches it.
- Option C: Incorrect, no horizontal change has been made to the function.
- Option D: Correct, the function [tex]\(g(x) = 7x^2\)[/tex] represents a vertical stretch by a factor of 7.
Therefore, the correct answer is:
D. The graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] vertically stretched by a factor of 7.
