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Questions and Answers
What is the correct $y$-intercept of the function $f(x)=(3x-2)(x+2)^2$?
What is the correct $y$-intercept of the function $f(x)=(3x-2)(x+2)^2$?
Which of the following accurately describes the $x$-intercepts of the function $f(x)$?
Which of the following accurately describes the $x$-intercepts of the function $f(x)$?
What is the behavior of the graph of $f(x)$ at $x = -2$?
What is the behavior of the graph of $f(x)$ at $x = -2$?
As $x$ approaches $+ ext{infinity}$, what does $f(x)$ approach?
As $x$ approaches $+ ext{infinity}$, what does $f(x)$ approach?
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Which statement is true regarding the multiplicity of the zero $ rac{2}{3}$ in the function $f(x)$?
Which statement is true regarding the multiplicity of the zero $ rac{2}{3}$ in the function $f(x)$?
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To find the end behavior of the function, which of the following must be determined?
To find the end behavior of the function, which of the following must be determined?
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What does the end behavior of a polynomial function indicate?
What does the end behavior of a polynomial function indicate?
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Analyzing the function $f(x)=(3x-2)(x+2)^2$, which of the following is a characteristic of the graph?
Analyzing the function $f(x)=(3x-2)(x+2)^2$, which of the following is a characteristic of the graph?
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What is the leading term of the polynomial function f(x)?
What is the leading term of the polynomial function f(x)?
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What happens to f(x) as x approaches negative infinity?
What happens to f(x) as x approaches negative infinity?
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Which x-coordinate represents a point where the graph of f(x) touches the x-axis?
Which x-coordinate represents a point where the graph of f(x) touches the x-axis?
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Where is the y-intercept of the function f(x) located?
Where is the y-intercept of the function f(x) located?
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What can be concluded about the interval where f(x) is positive?
What can be concluded about the interval where f(x) is positive?
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What indicates that the zero at x = 2/3 is of odd multiplicity?
What indicates that the zero at x = 2/3 is of odd multiplicity?
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What is the end behavior of the function f(x) as x approaches positive infinity?
What is the end behavior of the function f(x) as x approaches positive infinity?
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Identify the degree of the polynomial f(x).
Identify the degree of the polynomial f(x).
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Which statement is true regarding the overall shape of the graph of f(x)?
Which statement is true regarding the overall shape of the graph of f(x)?
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What can be determined about the section of the graph between the x-intercepts at -2 and 2/3?
What can be determined about the section of the graph between the x-intercepts at -2 and 2/3?
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Study Notes
End Behavior of Polynomials
- End behavior describes how a function's graph behaves at the far left and far right of the x-axis.
- This is determined by examining the behavior of the function as x approaches positive infinity and as x approaches negative infinity.
Zeros of Polynomials
- Zeros of a function (x-intercepts) are where the graph intersects the x-axis.
- If a zero has odd multiplicity, the graph crosses the x-axis at that point.
- If a zero has even multiplicity, the graph touches the x-axis but does not cross it at that point.
Analyzing Polynomial Functions
- To analyze a polynomial function, you need to find:
- The y-intercept: Find f(0)
- x-intercepts: Solve f(x) = 0. This often involves the zero product property, where if a product is 0, at least one of the factors equals 0.
- End behavior: Find the leading term of the polynomial in standard form (highest power of x).
- The end behavior of the function will mirror the end behavior of the leading term.
- If the degree (highest power) is odd and the leading coefficient is positive, the graph will increase from left to right (positive on the right, negative on left).
Example Analysis of f(x) = (3x - 2)(x + 2)²
- y-intercept: f(0) = -8 (Point: (0, -8))
- x-intercepts: Solving (3x-2)(x+2)² = 0 gives x = 2/3 (multiplicity 1) and x = -2 (multiplicity 2). The corresponding points are (2/3, 0) and (-2, 0).
- End behavior: The leading term in standard form is 3x³. Since the degree is odd and the leading coefficient is positive, the graph increases from left to right (negative infinity to negative infinity).
Sketching the Graph
- The graph touches the x-axis at (-2, 0) and crosses at (2/3, 0).
- The end behavior is similar to y = x³
- Plot the y-intercept.
Positive and Negative Intervals
- Based on the sketch of the graph determine intervals where the function is above (positive) or below (negative) the x-axis.
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Description
This quiz focuses on understanding the end behavior and zeros of polynomial functions. You will learn how to analyze a polynomial's graph by identifying its intercepts and determining its behavior at the extremes of the x-axis. Test your knowledge on these critical concepts in polynomial analysis.
