Question

How do I get the information needed to graph this? Help please

Answer 1

See attachment for graph.

Explanation:

Given polynomial function:

`f(x)=x^3-3x^2`

Axis intercepts

The zeros of a function are the points at which the curve crosses the x-axis. To find the zeros, factor the function and set it to zero:

`\implies f(x)=x^2(x-3)`

Therefore:

`\implies x^2=0 \implies x=0\: \textsf{ with multiplicity 2}`

`\implies (x-3)=0 \implies x=3`

Therefore, the curve intersects the x-axis at x = 3 and has a turning point at (0, 0) since a zero with even multiplicity touches the x-axis and bounces off of the axis.

Note: Since the curve touches the x-axis at (0, 0), the y-intercept is also (0, 0).

End behavior

The end behavior of a function is the behavior the graph of f(x) as x approaches -∞ or +∞.

The degree and the leading coefficient of a polynomial function determine the end behavior of the graph.

As the degree of the given function is odd and the leading coefficient is positive, the end behaviors are:

• 
  `\textsf{As } x \rightarrow - \infty, \:f(x) \rightarrow - \infty`

• 
  `\textsf{As } x \rightarrow \infty, \:f(x) \rightarrow \infty`

Turning points

We have already determined that there is a turning point at (0, 0).

To find the other turning point, differentiate the function:

`\boxed{\begin{minipage}{5 cm}\underline{Differentiating $x^n$}\\\\If $y=x^n$, then $\frac{\text{d}y}{\text{d}x}=xn^(n-1)$\\\end{minipage}}`

`\implies f'(x)=3\cdot x^(3-1)-2 \cdot 3x^(2-1)`

`\implies f'(x)=3x^2-6x`

Set the differentiated function to zero and solve for x:

`\implies 3x^2-6x=0`

`\implies 3x(x-2)=0`

`\implies 3x=0 \implies x=0`

`\implies (x-2)=0 \implies x=2`

Therefore, there are turning points at x = 0 and x = 2.

Substitute x = 2 into the function to find the y-value of this turning point.

`\implies f(2)=(2)^3-3(2)^2=-4`

Therefore, the turning points are at (0, 0) and (2, -4).

In Summary

• Cubic curve.

• End behaviors:
  
  `\textsf{As } x \rightarrow - \infty, \:f(x) \rightarrow - \infty`
  
  `\textsf{As } x \rightarrow \infty, \:f(x) \rightarrow \infty`

• Turning points at (0, 0) and (2, -4).

• Crosses the x-axis at (3, 0).