Graphing Polynomial Functions. Finding the End Behavior of a function Degree Leading Coefficient Graph Comparison End Behavior As x  – , Rise right.

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Graphing Polynomial Functions

Finding the End Behavior of a function Degree Leading Coefficient Graph Comparison End Behavior As x  – , Rise right Rise left Fall right Fall left Rise right Fall left Fall right Rise left y = x 2 y = –x 2 y = x 3 y = –x 3 Positive Negative Positive Negative Even Odd f(x)  ++ As x  + ,f(x)  As x  + ,f(x)  As x  – ,f(x)  As x  + , As x  – , f(x)  ++ –– –– ++ –– –– ++

Determine the end behavior of the function. f(x) = 4x 7 + 5x Degree:________ Leading Coefficient: ___________ Graph Comparison:________ End Behavior As x  – ,f(x)  –– Odd Positive y = x 3 As x  + ,f(x)  ++ Rises right Falls left 1

Leading Coefficient: ___________ f(x) = –7x 6 + 2x 2 – 3x Degree:________ Graph Comparison:________ End Behavior As x  – ,f(x)  –– Even Negative y = –x 2 As x  + ,f(x)  –– Falls right Falls left Determine the end behavior of the function. 2

Leading Coefficient: ___________ f(x) = –2x 5 – x 3 + 6x Degree:________ Graph Comparison:________ End Behavior As x  – ,f(x)  ++ Odd Negative y = –x 3 As x  + ,f(x)  –– Falls right Rises left Determine the end behavior of the function. 3

Left Side Graph Comparison 4 Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. Right Side FallsRises y = x 3 Degree:________ Odd Leading Coefficient: ___________ Positive

Left Side Graph Comparison 4 Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. Right Side Rises Degree:________ Even Leading Coefficient: ___________ Positive y = x 2

Left Side Graph Comparison 4 Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. Right Side RisesFalls Degree:________ Odd Leading Coefficient: ___________ Negative y = –x 3

The multiplicity of root r is the number of times that x – r is a factor of P(x). Multiplicity The graph of P(x) touches the x-axis, but does not cross it. (i.e. U-turn) Odd Multiplicity The graph of P(x) crosses the x-axis. Even Multiplicity Root a has odd multiplicity Root b has even multiplicity

Find the following on chart for problems #5, #6, #7 Zeros (Solutions) x-intercepts Multiplicity of zeros Explain what occurs on graph at each x-intercept

Find the zeros and x-intercepts for the function. Let P(x) = = (x – 1) 2 (x – 3) x – 1 = 0, x – 3 = 0 Let each factor = 0 Solve for x x = 1x = 3 x-intercepts (1,0) zeros (3,0)

Find the multiplicity for each zero, and explain what occurs at each x-intercept. 5 x = 1x = 3 Zeros Multiplicity At x = 12 The graph touches the x-axis and turns around at the x-intercept Multiplicity At x = 31 The graph crosses the x-axis at the x-intercept.

Find the zeros and x-intercepts for the function. Let P(x) = 0 6 x + 2 = 0, x – 3 = 0 Let each factor = 0 Solve for x x = –2x = 3 x-intercepts (–2,0) zeros (3,0)

Find the multiplicity for each zero, and explain what occurs at each x-intercept. 6 x = –2x = 3 Zeros Multiplicity At x = –22 The graph touches the x-axis and turns around at the x-intercept Multiplicity At x = 32 The graph touches the x-axis and turns around at the x-intercept

Find the zeros and x-intercepts for the function. Let P(x) = = x(x – 3)(x + 2), x – 3 = 0, x + 2 = 0 Let each factor = 0 Solve for x x = 3x = –2 x-intercepts (3,0) zeros (–2,0) x = 0 (0,0)

Find the multiplicity for each zero, and explain what occurs at each x-intercept. 7 Zeros Multiplicity At x = 01 The graph crosses the x-axis at the x-intercept. Multiplicity At x = 31 The graph crosses the x-axis at the x-intercept. x = 3x = –2x = 0 Multiplicity At x = –21 The graph crosses the x-axis at the x-intercept.

Degree Leading Coefficient Graph Comparison End Behavior Maximum Number of Turning Points Find the following on chart for problems #5, #6, #7

Find the following. 5 Degree:________ Leading Coefficient: ___________ Graph Comparison:______ End Behavior As x  – ,f(x)  –– Odd Positive y = x 3 As x  + ,f(x)  ++ Rises right Falls left = 3 P(x) = 1(x – 1) 2 (x – 3) 1 1 Maximum Number Of Turning Points = Degree – 1= 3 – 1= 2

Find the following. 6 Leading Coefficient: ___________ Degree:________ Graph Comparison:______ End Behavior As x  – ,f(x)  ++ Even Positive y = x 2 As x  + ,f(x)  ++ Rises right Rises left = 4 Maximum Number Of Turning Points = Degree – 1= 4 – 1= 3

Find the following. 7 Degree:________ Leading Coefficient: ___________ Graph Comparison:______ End Behavior As x  – ,f(x)  –– Odd Positive y = x 3 As x  + ,f(x)  ++ Rises right Falls left = 3 P(x) = 1x 1 (x – 3) 1 (x + 2) 1 1 P(x) = x(x – 3)(x + 2) Maximum Number Of Turning Points = Degree – 1= 3 – 1= 2

Find y-intercept Find Additional Points Plot Points Use Multiplicity Draw Graph Find the following on chart for problems #5, #6, #7

Find the y-intercept for the function. Substitute x = 0. y-intercept: (0,–3) 5 y = (x – 1) 2 (x – 3) y = (0 – 1) 2 (0 – 3) y = (–1) 2 (–3)= (1)(–3)= –3 Solve for y. Find additional points. Substitute x-value between zeros x = 1 and x = 3. Let x = 2: y = (2 – 1) 2 (2 – 3) y = (1) 2 (–1) y = (1)(–1)= –1 Point: (2,–1)

Draw graph. y-intercept: (0,–3) 5 y = (x – 1) 2 (x – 3) Point: (2,–1) x-intercepts (1,0) (3,0) Multiplicity 2 1 Falls left / Rises right End Behavior

Find the y-intercept for the function. Substitute x = 0. y-intercept: (0,3) 6 = 3 Solve for y.

6 Find additional points. Substitute x-value between zeros x = –2 and x = 3. Let x = 1: Point: (1,3) = 3

Draw graph. y-intercept: (0,3) 6 Point: (1,3) x-intercepts (–2,0) (3,0) Multiplicity 2 Rises left / Rises right End Behavior

Find the y-intercept for the function. Substitute x = 0. y-intercept: (0,0) 7 Solve for y. y = x(x – 3)(x + 2) y = 0(0 – 3)(0 + 2) y = 0(–3)(2) y = 0

7 y = x(x – 3)(x + 2) Find additional points. Substitute x-values between zeros x = –2, x = 0, and x = 3. Let x = –1: Point: (–1,4) y = –1(–1 – 3)(–1 + 2) y = –1(–4)(1)= 4 Let x = 1: Point: (1,–6) y = 1(1 – 3)(1 + 2) y = 1(–2)(3)= –6 Let x = 2: Point: (2,–8) y = 2(2 – 3)(2 + 2) y = 2(–1)(4)= –8

Draw graph. y-intercept: (0,0) 7 Points (–1,4) (1,–6) (2,–8) x-intercepts (0,0) (3,0) (–2,0) Multiplicity 1 Falls left / Rises right End Behavior y = x(x – 3)(x + 2)    