Analyzing Polynomial Functions

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Questions and Answers

What is the correct $y$-intercept of the function $f(x)=(3x-2)(x+2)^2$?

  • (0, -8) (correct)
  • (0, 0)
  • (0, -4)
  • (0, 2)

Which of the following accurately describes the $x$-intercepts of the function $f(x)$?

  • Two $x$-intercepts, both of multiplicity 2
  • Only one $x$-intercept of multiplicity 1
  • One $x$-intercept of multiplicity 2
  • Two $x$-intercepts, one of multiplicity 1 and another of multiplicity 2 (correct)

What is the behavior of the graph of $f(x)$ at $x = -2$?

  • The graph does not touch the x-axis
  • The graph touches the x-axis (correct)
  • The graph crosses the x-axis
  • The graph dips below the x-axis

As $x$ approaches $+ ext{infinity}$, what does $f(x)$ approach?

<p>A positive value (B)</p> Signup and view all the answers

Which statement is true regarding the multiplicity of the zero $ rac{2}{3}$ in the function $f(x)$?

<p>It is a zero of odd multiplicity. (B)</p> Signup and view all the answers

To find the end behavior of the function, which of the following must be determined?

<p>The leading coefficient and the degree of the polynomial (D)</p> Signup and view all the answers

What does the end behavior of a polynomial function indicate?

<p>It describes the function's output as $x$ approaches positive or negative infinity. (D)</p> Signup and view all the answers

Analyzing the function $f(x)=(3x-2)(x+2)^2$, which of the following is a characteristic of the graph?

<p>The function can have positive and negative intervals. (D)</p> Signup and view all the answers

What is the leading term of the polynomial function f(x)?

<p>3x^3 (D)</p> Signup and view all the answers

What happens to f(x) as x approaches negative infinity?

<p>f(x) approaches -∞ (B)</p> Signup and view all the answers

Which x-coordinate represents a point where the graph of f(x) touches the x-axis?

<p>(-2, 0) (C)</p> Signup and view all the answers

Where is the y-intercept of the function f(x) located?

<p>(0, -8) (D)</p> Signup and view all the answers

What can be concluded about the interval where f(x) is positive?

<p>For x &gt; 2/3 (A)</p> Signup and view all the answers

What indicates that the zero at x = 2/3 is of odd multiplicity?

<p>The graph crosses the x-axis at that point (D)</p> Signup and view all the answers

What is the end behavior of the function f(x) as x approaches positive infinity?

<p>f(x) approaches +∞ (D)</p> Signup and view all the answers

Identify the degree of the polynomial f(x).

<p>3 (D)</p> Signup and view all the answers

Which statement is true regarding the overall shape of the graph of f(x)?

<p>The ends extend upwards in both directions. (B)</p> Signup and view all the answers

What can be determined about the section of the graph between the x-intercepts at -2 and 2/3?

<p>It is decreasing throughout (C)</p> Signup and view all the answers

Flashcards

End behavior

The behavior of a function's graph at the extreme ends of the x-axis. It indicates where the graph goes as x approaches positive or negative infinity.

Zeros of a function

The values of x where the function's graph intersects the x-axis. They represent the points where the function equals zero.

Y-intercept

The point where the function's graph intersects the y-axis. It is found by evaluating the function at x=0.

Multiplicity of a zero

The number of times a zero appears as a factor in the function's factored form. It determines how the graph behaves at that zero.

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Graph behavior at zeros

If a zero has an odd multiplicity, the graph crosses the x-axis at that point. If it has an even multiplicity, the graph touches the x-axis at that point.

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Leading Term

The term with the highest power of the variable.

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Analyzing positive and negative intervals

The process of determining the intervals on the x-axis where the function's graph is above or below the x-axis. This can be done by analyzing the signs of the function's value in different intervals.

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Leading term

The leading term is the term with the highest power of x when the function is written in standard form. Its coefficient and exponent determine the end behavior of the function.

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Standard form of a polynomial

The standard form of a polynomial is written in descending order of exponents, with terms combined. It helps to identify the leading term and analyze the end behavior.

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Zero of a Polynomial

The x-value at which the graph of a function crosses the x-axis. It's the solution where the function equals zero.

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Degree of a Polynomial

The highest power of the variable in a polynomial.

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Leading Coefficient

The coefficient of the leading term in a polynomial.

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Even Multiplicity

A zero of even multiplicity touches the x-axis without crossing it. The graph flattens momentarily at the x-intercept.

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Odd Multiplicity

A zero of odd multiplicity crosses the x-axis. The graph changes direction at the x-intercept.

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Turning Point

A point on the graph where the curve changes direction (from increasing to decreasing or vice versa).

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Positive and Negative Intervals

The intervals on the x-axis where the function's values are positive (above the x-axis) or negative (below the x-axis).

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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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