To understand the transformations needed to change the graph of f(x) = 7x² into the graph of g(x) = 35x² + 5, we can break it down into two main components: vertical stretching/compressing and vertical shifting.
Firstly, we notice that the leading coefficient of g(x) is greater than that of f(x). Specifically:
f(x) = 7x²
g(x) = 35x² + 5
The coefficient increases from 7 to 35, which means the graph is vertically stretched by a factor of 5 (because 35 is 5 times 7). This transformation makes the parabola narrower compared to f(x).
Secondly, g(x) has a constant term of +5. This indicates a vertical shift of the entire graph upwards by 5 units. This means every point on the graph of f(x) is moved up 5 units to form the graph of g(x).
In summary, the transformations involved are:
- Vertical Stretch: The graph is stretched vertically by a factor of 5.
- Vertical Shift: The graph is shifted upwards by 5 units.
These transformations change f(x) = 7x² into g(x) = 35x² + 5.