# 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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