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NO LINKS!! Please help me with these graphs. (NOT Multiple choice)

a. Show the end behavior
b. Shape the graph near the near x-intercepts.

1. p(x)= 2/3(3x - 6)(x + 2) (x - 3)^2

2. p(x) = (x - 3)^2 (x + 2) (2x + 7)^2

3. p(x) = (x - 5)(x + 2) (x - 1) - 30

User Ali Fallah
by
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1 Answer

3 votes

Answer:


\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}


\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}


\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}

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:


p(x)= (2)/(3)(3x - 6)(x + 2)(x - 3)^2

Therefore:

  • Degree: 4 (even)
  • Leading coefficient: positive

End behaviors:


\textsf{As}\;\; x \rightarrow - \infty,\;\;f(x) \rightarrow + \infty


\textsf{As}\;\; x \rightarrow +\infty,\;\;f(x) \rightarrow + \infty

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:


p(x)=(x - 3)^2 (x + 2) (2x + 7)^2

Therefore:

  • Degree: 5 (odd)
  • Leading coefficient: positive

End behaviors:


\textsf{As}\;\; x \rightarrow - \infty,\;\;f(x) \rightarrow -\infty


\textsf{As}\;\; x \rightarrow +\infty,\;\;f(x) \rightarrow + \infty

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:


p(x)=(x - 5)(x + 2)(x - 1)-30

Therefore:

  • Degree: 3 (odd)
  • Leading coefficient: positive

End behaviors:


\textsf{As}\;\; x \rightarrow - \infty,\;\;f(x) \rightarrow -\infty


\textsf{As}\;\; x \rightarrow +\infty,\;\;f(x) \rightarrow + \infty

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
NO LINKS!! Please help me with these graphs. (NOT Multiple choice) a. Show the end-example-1
NO LINKS!! Please help me with these graphs. (NOT Multiple choice) a. Show the end-example-2
NO LINKS!! Please help me with these graphs. (NOT Multiple choice) a. Show the end-example-3
User Gerriet
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7.2k points