Question
Write a polynomial function that imitates the end behavior of the graph shown to the right. The dashed portion of the graph indicates that you should focus only on imitating the left and right behavior of the graph and can be flexible about what occurs between the left and right ends.
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Part 1
Which of the following polynomial functions imitates the end behavior of the graph shown above?
f left parenthesis x right parenthesis equals negative 9 x Superscript 5 Baseline plus 5 x Superscript 4 Baseline plus 9
f left parenthesis x right parenthesis equals negative 9 x Superscript 5 Baseline plus 5 x Superscript 4 Baseline plus 9
f left parenthesis x right parenthesis equals 9 x squared minus 7 x plus 10
f left parenthesis x right parenthesis equals 9 x squared minus 7 x plus 10
f left parenthesis x right parenthesis equals 3 x Superscript 5 Baseline plus 2 x Superscript 4 Baseline minus 7
f left parenthesis x right parenthesis equals 3 x Superscript 5 Baseline plus 2 x Superscript 4 Baseline minus 7
f left parenthesis x right parenthesis equals negative 9 x Superscript 6 Baseline plus 3 x plus 2
f left parenthesis x right parenthesis equals negative 9 x Superscript 6 Baseline plus 3 x plus 2
Question content area bottom
Part 1
Which of the following polynomial functions imitates the end behavior of the graph shown above?
f left parenthesis x right parenthesis equals negative 9 x Superscript 5 Baseline plus 5 x Superscript 4 Baseline plus 9
f left parenthesis x right parenthesis equals negative 9 x Superscript 5 Baseline plus 5 x Superscript 4 Baseline plus 9
f left parenthesis x right parenthesis equals 9 x squared minus 7 x plus 10
f left parenthesis x right parenthesis equals 9 x squared minus 7 x plus 10
f left parenthesis x right parenthesis equals 3 x Superscript 5 Baseline plus 2 x Superscript 4 Baseline minus 7
f left parenthesis x right parenthesis equals 3 x Superscript 5 Baseline plus 2 x Superscript 4 Baseline minus 7
f left parenthesis x right parenthesis equals negative 9 x Superscript 6 Baseline plus 3 x plus 2
f left parenthesis x right parenthesis equals negative 9 x Superscript 6 Baseline plus 3 x plus 2
Answers
Answered by
GPT-4o mini
To determine which polynomial function imitates the end behavior of the graph described, we need to analyze how the graph behaves as \( x \) approaches negative infinity and positive infinity.
**End behavior rules for polynomials:**
1. If the leading coefficient is positive and the degree is even, the graph goes up on both ends \((+\infty\) as \(x \to -\infty\) and \(+\infty\) as \(x \to +\infty\)).
2. If the leading coefficient is positive and the degree is odd, the graph goes down on the left and up on the right \((-\infty\) as \(x \to -\infty\) and \(+\infty\) as \(x \to +\infty\)).
3. If the leading coefficient is negative and the degree is even, the graph goes down on both ends \((-\infty\) as \(x \to -\infty\) and \(-\infty\) as \(x \to +\infty\)).
4. If the leading coefficient is negative and the degree is odd, the graph goes up on the left and down on the right \((+\infty\) as \(x \to -\infty\) and \(-\infty\) as \(x \to +\infty\)).
**Given choices:**
1. \( f(x) = -9x^5 + 5x^4 + 9 \) (degree 5, leading coefficient negative)
2. \( f(x) = 9x^2 - 7x + 10 \) (degree 2, leading coefficient positive)
3. \( f(x) = 3x^5 + 2x^4 - 7 \) (degree 5, leading coefficient positive)
4. \( f(x) = -9x^6 + 3x + 2 \) (degree 6, leading coefficient negative)
**Analyzing the end behavior:**
- For \( -9x^5 + 5x^4 + 9 \): Since it has a degree of 5 (odd) and a negative leading coefficient, the graph will imitate the end behavior of \((+\infty\) as \(x \to -\infty\) and \(-\infty\) as \(x \to +\infty)\).
- For \( 9x^2 - 7x + 10 \): Since it has a degree of 2 (even) and a positive leading coefficient, it will go to \(-\infty\) as \(x \to -\infty\) and \(+\infty\) as \(x \to +\infty\).
- For \( 3x^5 + 2x^4 - 7 \): This has a degree of 5 (odd) and a positive leading coefficient, meaning it’s going to have an end behavior of \(-\infty\) as \(x \to -\infty\) and \(+\infty\) as \(x \to +\infty\).
- For \( -9x^6 + 3x + 2 \): This is a degree of 6 (even) with a negative leading coefficient, which means it will head to \(-\infty\) on both ends.
Since you focus mainly on the left and right behavior, if your description indicates that the graph goes to \((+\infty\) on the left and \(-\infty\) on the right\), only the first option fits that description.
Thus, the polynomial function that imitates the end behavior of the graph is: **\( f(x) = -9x^5 + 5x^4 + 9 \)**.
**End behavior rules for polynomials:**
1. If the leading coefficient is positive and the degree is even, the graph goes up on both ends \((+\infty\) as \(x \to -\infty\) and \(+\infty\) as \(x \to +\infty\)).
2. If the leading coefficient is positive and the degree is odd, the graph goes down on the left and up on the right \((-\infty\) as \(x \to -\infty\) and \(+\infty\) as \(x \to +\infty\)).
3. If the leading coefficient is negative and the degree is even, the graph goes down on both ends \((-\infty\) as \(x \to -\infty\) and \(-\infty\) as \(x \to +\infty\)).
4. If the leading coefficient is negative and the degree is odd, the graph goes up on the left and down on the right \((+\infty\) as \(x \to -\infty\) and \(-\infty\) as \(x \to +\infty\)).
**Given choices:**
1. \( f(x) = -9x^5 + 5x^4 + 9 \) (degree 5, leading coefficient negative)
2. \( f(x) = 9x^2 - 7x + 10 \) (degree 2, leading coefficient positive)
3. \( f(x) = 3x^5 + 2x^4 - 7 \) (degree 5, leading coefficient positive)
4. \( f(x) = -9x^6 + 3x + 2 \) (degree 6, leading coefficient negative)
**Analyzing the end behavior:**
- For \( -9x^5 + 5x^4 + 9 \): Since it has a degree of 5 (odd) and a negative leading coefficient, the graph will imitate the end behavior of \((+\infty\) as \(x \to -\infty\) and \(-\infty\) as \(x \to +\infty)\).
- For \( 9x^2 - 7x + 10 \): Since it has a degree of 2 (even) and a positive leading coefficient, it will go to \(-\infty\) as \(x \to -\infty\) and \(+\infty\) as \(x \to +\infty\).
- For \( 3x^5 + 2x^4 - 7 \): This has a degree of 5 (odd) and a positive leading coefficient, meaning it’s going to have an end behavior of \(-\infty\) as \(x \to -\infty\) and \(+\infty\) as \(x \to +\infty\).
- For \( -9x^6 + 3x + 2 \): This is a degree of 6 (even) with a negative leading coefficient, which means it will head to \(-\infty\) on both ends.
Since you focus mainly on the left and right behavior, if your description indicates that the graph goes to \((+\infty\) on the left and \(-\infty\) on the right\), only the first option fits that description.
Thus, the polynomial function that imitates the end behavior of the graph is: **\( f(x) = -9x^5 + 5x^4 + 9 \)**.