Transformations from f(x) to g(x)
The graphs of the two functions, f(x) = 7x² and g(x) = 35x² + 5, can be understood in terms of several key transformations:
1. Vertical Stretch
The first transformation involves a vertical stretch of the graph. The coefficient of x² in f(x) is 7, while in g(x) it is 35. This means that g(x) is a vertical stretch of f(x).
Specifically, multiplying by 5 (since 35 is 5 times 7) means that the graph of g(x) will be stretched vertically, making it thinner and steeper compared to f(x).
2. Vertical Shift
The second transformation is a vertical shift. In the equation for g(x), you can see the +5 at the end. This means that the entire graph of f(x) is shifted upward by 5 units.
So, every point on the graph of f(x) will move up 5 units in the graph of g(x).
Summary
In summary, to transform f(x) = 7x² to g(x) = 35x² + 5, you need to:
- Apply a vertical stretch by a factor of 5.
- Shift the graph vertically upwards by 5 units.