Describe the transformation of y=5(x-1)^2 from the parent function (y=x^2)
the 5 is a scale factor, so the graph is stretched by a factor of 5 vertically.
(x-1) shifts the graph to the right one unit.
Thank you!
To describe the transformation of the function y=5(x-1)^2 from the parent function y=x^2, we need to analyze the changes that occur to the original function.
The parent function y=x^2 is a simple quadratic function that is symmetric about the y-axis and passes through the origin (0, 0).
Now, let's look at the transformation applied to the parent function in y=5(x-1)^2:
1. Horizontal Translation:
The expression "(x-1)" inside the parentheses represents a horizontal translation. The term "x-1" shifts the entire graph one unit to the right. This means that all the x-values that would originally be at, for example, x=0 on the parent function, are now at x=1 on the transformed function.
2. Vertical Stretch:
The coefficient "5" outside the parentheses represents a vertical stretch or compression. In this case, because the coefficient "5" is greater than 1, the graph gets stretched vertically. The y-values on the transformed function will be five times larger than the equivalent y-values on the parent function. This leads to a steeper curve.
3. No Vertical Translation:
The expression inside the parentheses, "(x-1)^2", does not introduce any vertical translation. It only modifies the shape of the parent function, not its vertical position.
To summarize, the transformation of the parent function y=x^2 to y=5(x-1)^2 includes a horizontal shift to the right by one unit and a vertical stretch by a factor of 5. The overall shape of the graph remains the same, just taller and shifted horizontally.