Answer:
[tex]\begin{aligned}\textsf{1.} \quad \textsf{As}\;\; &x \rightarrow - \infty,\;\;f(x) \rightarrow + \infty\\ \textsf{As}\;\; &x \rightarrow +\infty,\;\;f(x) \rightarrow + \infty \end{aligned}[/tex]
[tex]\begin{aligned}\textsf{2.} \quad \textsf{As}\;\; &x \rightarrow - \infty,\;\;f(x) \rightarrow - \infty\\ \textsf{As}\;\; &x \rightarrow +\infty,\;\;f(x) \rightarrow + \infty \end{aligned}[/tex]
[tex]\begin{aligned}\textsf{3.} \quad \textsf{As}\;\; &x \rightarrow - \infty,\;\;f(x) \rightarrow - \infty\\ \textsf{As}\;\; &x \rightarrow +\infty,\;\;f(x) \rightarrow + \infty \end{aligned}[/tex]
Step-by-step explanation:
Root Multiplicity
Odd multiplicity → the graph will cross the x-axis at the root.
Even multiplicity → the graph will touch the x-axis at the root (but will not cross the x-axis).
Question 1
Given function:
[tex]p(x)= \dfrac{2}{3}(3x - 6)(x + 2)(x - 3)^2[/tex]
Therefore:
- Degree: 4 (even)
- Leading coefficient: positive
End behaviors:
[tex]\textsf{As}\;\; x \rightarrow - \infty,\;\;f(x) \rightarrow + \infty[/tex]
[tex]\textsf{As}\;\; x \rightarrow +\infty,\;\;f(x) \rightarrow + \infty[/tex]
The x-intercepts of a function are when f(x)=0.
Therefore, the x-intercepts are:
- x = -2 with multiplicity 1
- x = 2 with multiplicity 1
- x = 3 with multiplicity 2
Therefore, the graph of the given function has 3 turning points.
- Begins in quadrant II
- Crosses the x-axis at x = -2
- Crosses the x-axis at x = 2
- Touches the x-axis at x = 3
- Ends in quadrant I
Question 2
Given function:
[tex]p(x)=(x - 3)^2 (x + 2) (2x + 7)^2[/tex]
Therefore:
- Degree: 5 (odd)
- Leading coefficient: positive
End behaviors:
[tex]\textsf{As}\;\; x \rightarrow - \infty,\;\;f(x) \rightarrow -\infty[/tex]
[tex]\textsf{As}\;\; x \rightarrow +\infty,\;\;f(x) \rightarrow + \infty[/tex]
The x-intercepts of a function are when f(x)=0. Therefore, the x-intercepts are:
- x = -3.5 with multiplicity 2
- x = -1 with multiplicity 1
- x = 3 with multiplicity 2
Therefore, the graph of the given function has 4 turning points.
- Begins in quadrant III
- Touches the x-axis at x = -3.5
- Crosses the x-axis at x = -2
- Touches the x-axis at x = 3
- Ends in quadrant I
Question 3
Given function:
[tex]p(x)=(x - 5)(x + 2)(x - 1)-30[/tex]
Therefore:
- Degree: 3 (odd)
- Leading coefficient: positive
End behaviors:
[tex]\textsf{As}\;\; x \rightarrow - \infty,\;\;f(x) \rightarrow -\infty[/tex]
[tex]\textsf{As}\;\; x \rightarrow +\infty,\;\;f(x) \rightarrow + \infty[/tex]
The graph has 2 turning points.
- Begins in quadrant III
- Turning points at x ≈ -0.7 and x ≈ 3.4
- Crosses the x-axis at x ≈ 5.8
- Ends in quadrant I
