When p is an odd positive integer, then the graph of the power function f(x)=x^p is similar to the graph of y=??

y = x^3

All of the odd powers look basically the same as x^3. Play around at wolframalpha.com to see.

When p is an odd positive integer, the graph of the power function f(x) = x^p is similar to the graph of a roller coaster ride. Why? Well, just like a roller coaster, the graph of f(x) = x^p has its ups and downs! It starts from the origin (0,0), rises to positive values, then descends back down to negative values as x moves further away from zero. It's a wild and thrilling ride, just like a roller coaster! So, buckle up and enjoy the mathematical adrenaline, my friend!

When p is an odd positive integer, the graph of the power function f(x) = x^p is similar to the graph of y = x.

To determine the graph of the power function f(x) = x^p when p is an odd positive integer, we need to consider the behavior of the function for various values of x.

When x is a positive value, raising it to an odd power preserves the sign. For example, if we take x = 2, then 2^1 = 2, 2^3 = 8, 2^5 = 32, and so on. As you can see, in each case, the result is positive. Similarly, if we take x = -2, then (-2)^1 = -2, (-2)^3 = -8, (-2)^5 = -32, and so on. The result is negative in each case.

Based on this behavior, when x is positive, the graph of f(x) = x^p starts in quadrant I and gradually curves upward toward positive infinity as x increases. On the other hand, when x is negative, the graph starts in quadrant III and curves downward toward negative infinity as x decreases.

To summarize, when p is an odd positive integer, the graph of f(x) = x^p is similar to the graph of y = x, but with a steeper curve. Both functions start in quadrant I and curve upwards towards positive infinity for positive values of x. However, for negative values of x, they start in quadrant III and curve downwards towards negative infinity.

Note that this explanation is generalized for odd positive p-values and assumes that x can take any real number values. If p is a fractional or negative number, different rules would apply.