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Quadratic Equation Graph Sketching and End Behavior Analysis

Learn how to sketch the graph of a quadratic equation using its vertex and intercepts, determine if it has a minimum or maximum value, and analyze the end behavior based on the leading coefficient and degree. Also, understand how to find the zeros of a polynomial function and identify their multiplicities.

elizabethwjohnson
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Quadratic Equation Graph Sketching and End Behavior Analysis

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  1. Warm-Up 4/25/13 • Answers: • See graph to the right. • (1,-10) 1. Use the vertex and intercepts to sketch the graph of this quadratic equation. f(x)= x2+3x-10 2. Decide if the graph has a minimum value or maximum value. Then find the min or max and determine where it occurs. f(x) = -3x2 + 6x -13 Homework: Pgs 322-323 (2-32 even)

  2. Lesson 3.2 Polynomial Functions and their Graphs

  3. Smooth, Continuous Graphs • Smooth means the graph contains only rounded curves. • Continuous means that graphs have no breaks in them and can be drawn without lifting your pencil.

  4. Sharp, Discontinuous Graphs • Sharp corners • Discontinuous Graphs mean a break in the graph and you have to lift your pencil.

  5. End Behaviors of Polynomial Graphs • Definition: The behavior of a graph of a function to the far left or the far right. • The sign of the leading coefficient, an, and the degree, n, of the polynomial function reveal its end behavior. • In terms of the end behavior, only the term of highest degree counts, as summarized by the Leading Coefficient Test.

  6. TIP: Odd degree polynomial functions have graphs with opposite behavior at each end.

  7. TIP: Even degree functions have graphs with the same behavior at each end.

  8. Using the Leading Coefficient Test f(x) = x3+ 3x2 – x - 3 The degree of the polynomial, 3 is odd. • f(x) = x3+ 3x2 – x - 3 • Because the degree of the polynomial is odd, the graph has opposite behaviors at both ends. • Because the leading coefficient is 1, the graph falls to the left and rises to the right. The leading coefficient, 1, is positive.

  9. Identify the end behavior of this polynomial.Ex. 2: f(x) =-49x3+806x2+3776x+2503 • Answer: The degree of f is 3, which is odd. Odd-degree functions have graphs with opposite behaviors at each end. • The leading coefficient is -49, is negative, therefore, the graph rises to the left and falls to the right.

  10. Ex. 3: f(x)= -x4+8x3 +4x2 +2 graph on your calculator with windows set at [-8,8,1] by [-10,10,1] • The degree of f is 4, which is even. Even degree polynomial functions have graphs with the same behaviors at each end. • The leading coefficient , -1, is negative, therefore, the graph should fall to the left and fall to the right. Does the graph show the end behavior of the function? The graph is falling to the left, but it is not falling to the right. Therefore, the graph is not complete enough to show end behavior. You will need to adjust your window max and mins. A more complete graph of the function is shown if you use a larger viewing rectangle. Try [-10,10,1] and [-1000, 750,250] What happens? Answer: The graph shows the end behavior if you use a larger viewing area.

  11. Example 4: Finding Zeros of a Polynomial Function • Find all zeros of f(x) = x3+3x2-x-3 x3+3x2-x-3=0 Step 1: Set function equal to 0. Step 2: Factor out GCF. x2(x+3) -1(x+3) =0 Step 3: Factor by grouping. (X2-1) (x+3)=0 X2-1=0 and x+3 =0 Step 4: Set each factor =0. Step 5: Solve for x. X = -3, x= 1, x= -1

  12. The x values tell you where the graph crosses the x axis.

  13. Example 5: Finding Zeros of a Polynomial Function • f(x) = -x4 +4x3 – 4x2 Factor out GCF. -x2(x2-4x+4) = 0 -x2(x-2)2=0 Factor completely. Set factors =0. -x2=0 and (x-2)2=0 Solve for x. X = 0 and x = 2

  14. Multiplicities of Zeros Even Multiplicity – when a zero/root has an even exponent, then the graph touches the x-axis and turns around at that zero/root. Odd Multiplicity – when a zero/root has an odd exponent, then the graph crosses the x-axis at that zero/root.

  15. Finding Zeros and their Multiplicities: Find the zeros of f(x) = (x+1)(2x-3)2 and give the multiplicity of each zero. • The root for x+ 1=0 is x = -1, and it has an odd multiplicity which means the graph crosses the x-axis at -1. • The root 2x-3=0 is x = 3/2, and it has an even multiplicity which means the graph touches the x-axis at 3/2 and turns around.

  16. Picture of Graph:

  17. You Try: Find the zeros of f(x) = -4(x+1/2)2(x+5)3 and give the multiplicity of each zero. State whether the graph crosses the x-axis or touches that x-axis and turns around at each zero. Answer: -1/2 with multiplicity 2(which is even) and 5 with multiplicity 3 (which is odd) , touches and turns at -1/2 and crosses at 5

  18. Summary: • What is meant by the end behavior of a polynomial function? • How do you determine the multiplicity of a function?

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