Given the base function f(x)=x^4, describe in point form how each parameter of the function y=3f(-2x-8)+5 transforms the graph of f(x).
Is it vertical stretch by a factor of3, horizontal compression by 1/2, reflection in y axis and translation 5 units up, 4 units left?
correct
To understand how each parameter of the function y=3f(-2x-8)+5 transforms the graph of f(x), let's break down the equation and consider each parameter one by one:
1. The function f(x)=x^4 represents the base function. This is a simple upward-opening parabola.
2. The parameter -2 in the expression -2x determines the horizontal stretch or compression of the graph. When this value is less than 1, like -2, it compresses the graph horizontally. In this case, it compresses the graph by a factor of 2.
3. The parameter -8 in the expression -2x-8 determines the horizontal shift of the graph. When this value is negative, like -8, the graph is shifted to the right by 8 units. This means that every point on the graph is moved 8 units to the right.
4. The parameter 3 in the expression 3f(-2x-8) determines the vertical stretch or compression of the graph. When this value is greater than 1, like 3, it stretches the graph vertically. In this case, it stretches the graph by a factor of 3.
5. Finally, the parameter 5 in the expression 3f(-2x-8)+5 determines the vertical shift of the graph. When this value is positive, like 5, the graph is shifted upward by 5 units. This means that every point on the graph is moved 5 units upward.
So, to summarize in point form:
- The parameter -2 compresses the graph horizontally by a factor of 2.
- The parameter -8 shifts the graph to the right by 8 units.
- The parameter 3 stretches the graph vertically by a factor of 3.
- The parameter 5 shifts the graph upward by 5 units.