Graph of y = x^(p/q)
Graphs of $y = x^n$ where n is a rational number (a fraction) $n = \dfrac{p}{q}$
These are variations on the shape of the square root or cube root graph.
Even Denominator, p/q < 1
If the denominator of the exponent is even, and the exponent is between 0 and 1
- the graph will have a similar shape to the square root graph.
- as x gets larger, the root dominates
The domain will be $\{ x: x \geqslant 0 \}$
All graphs will begin at (0, 0) and pass through (1, 1)
- Graphs with smaller values of p/q will rise steeply between 0 and 1
- Then will increase with a shallow gradient to infinity
- Graphs with larger values of p/q will rise slowly from 0 to 1
- Then will increase with a slowly decreasing gradient to infinity
Even Denominator, p/q > 1
If the denominator of the exponent is even, and the exponent is greater than 1
- the graph will have a similar shape to the right half of a parabola.
- as x gets larger, the numerator dominates.
The domain will be $\{ x: x \geqslant 0 \}$
All graphs will begin at (0, 0) and pass through (1, 1)
- Graphs with smaller values of p/q will rise steeply between 0 and 1
- Then will increase with a shallow gradient to infinity
- Graphs with larger values of p/q will rise slowly from 0 to 1
- Then will increase with a steeper gradient to infinity
Odd Denominator, Odd numerator, p/q < 1
If the denominator of the exponent is odd, the numerator is odd and the exponent is between 0 and 1
- the graph will have a similar shape to the cube root graph.
- as x gets larger, the root dominates
The domain will be $\{ x: x \in R \}$
All graphs will pass through (-1, -1) then (0, 0) then (1, 1)
- At (0, 0) there will be a vertical point of inflection
- the derivative will be undefined at (0, 0)
- Graphs with smaller values of p/q will rise steeply between 0 and 1
- Then will increase with a shallow gradient to infinity
- Graphs with larger values of p/q will rise slowly from 0 to 1
- Then will increase with a slowly decreasing gradient to infinity
Odd Denominator, Even numerator, p/q < 1
If the denominator of the exponent is odd, the numerator is even and the exponent is between 0 and 1
- the right part (positive) of the graph will have a similar shape to the cube root graph.
- as x gets larger, the root dominates
- the left part (negative) gets reflected up across the x-axis
- because the even numerator squares the negative values
The domain will be $\{ x: x \in R \}$
Notice that at (0, 0)
- there is a point in the graph (correct term is a cusp)
- the derivative is not defined at (0, 0) for these graphs
- the graphs will pass through (-1, 1) then (0, 0) then (1, 1)
Odd Denominator, p/q > 1
If the denominator of the exponent is odd, and the exponent is greater than 1
- the graph will have a similar shape to the cubic graph.
- as x gets larger, the numerator dominates
Again the shape will depend on the numerator
- graphs with an odd numerator and an odd denominator will resemble the cubic graph
- graphs with an even numerator and an odd denominator will have a cusp at (0, 0)
- the negative portion of the cubic graph will be reflected up across the x-axis
The domain for all of them will be $\{ x: x \in R \}$
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