Power Function: Exercise: Recognize and analyze the graph of the power function as the exponent "n" and the coefficient "a" vary.

1. Consider a positive exponent power function
A) Describe the graph of the function when n is even, for this you must attach an image of the graph of the function that meets these conditions.
B) Describe the graph of the function when n is odd, for this you must attach an image of the graph of the function that meets these conditions.
C) Describe what happened when the coefficient a is positive and also when it is negative. Is there any change in the graph?
D) Describe what happened when the coefficient a increases or decreases. What happens to the graph of the function?

2. Consider a negative exponent power function
A) Describe the graph of the function when n is even, for this you must attach an image of the graph of the function that meets these conditions.
B) Describe the graph of the function when n is odd, for this you must attach an image of the graph of the function that meets these conditions.
C) Describe what happened when the coefficient a is positive and also when it is negative. Is there any change in the graph?
D) Describe what happened when the coefficient a increases or decreases. What happens to the graph of the function?

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Explanation:

For this purpose, we define your power function as ...

  f(x) = a·x^n

Most of the description here applies to any function. A "power function" is a special case of a polynomial function in that it only has one term. The parity of the exponent (even, odd) tells you whether the function is even or odd. As with any even or odd function, there will be a certain symmetry:

  f(-x) = f(x) . . . even function; symmetrical about the y-axis

  f(-x) = -f(x) . . . odd function; symmetrical about the origin

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The coefficient of the function is the vertical scale factor. As with the vertical scale factor of any function, when it is negative, the function graph is reflected in the x-axis. When it increases in magnitude, points on the graph move farther from the x-axis; closer when it decreases.

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A) The red graphs in the first attachment are graphs of f(x) = ax^2 for different values of 'a'. As described above, the graph is symmetrical about the y-axis.

B) The black graphs in the first attachment are graphs of f(x) = ax^3 for different values of 'a'. As described above, the graph is symmetrical about the origin.

C) As described above, when the coefficient is negative, the graph of the parent function is reflected across the x-axis. The dashed-line graphs in the first attachment are graphs of the function with a negative coefficient.

D) As described above, the coefficient is the vertical scale factor. Changing it stretches or compresses the graph vertically. The dotted red graph in the first attachment shows a vertical expansion; the dashed black graph shows a vertical compression (reduction in the coefficient magnitude).

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The second attachment shows graphs with negative exponents. Otherwise, the graphs are scaled and reflected the same as they were in Part 1.

A) As described above, the graph is symmetrical about the y-axis.

B) As described above, the graph is symmetrical about the origin.

C) As described above, a negative coefficient causes the graph to be reflected across the x-axis.

D) As described above, the magnitude of the coefficient affects the distance of the graph from the x-axis. Increasing magnitude increases the distance; decreasing magnitude decreases the distance (proportionally).

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step-by-step explanation: