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The Second-Degree Polynomial Function

Learn about second-degree polynomial functions, standard and general forms of quadratic functions, transformations, properties of graphs, and how to convert between different forms of quadratic functions.

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The Second-Degree Polynomial Function

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  1. The Second-Degree Polynomial Function Quadratic Function

  2. y f(x)=x^2 5 x -8 -6 -4 -2 2 4 6 8 -5 The Basic Quadratic Function The graph of a second-degree polynomial (quadratic function) is presented as a parabola

  3. The Basic Quadratic Function • What does parameter “a” do? • Changes the vertical scale • Stretch if a1 (narrows the opening) • Shrink if 0a1 (widens the opening) • a0 (positive) the graph opens upward “smiles” • a0 (negative) the graph opens downward “frowns”

  4. The Basic Quadratic Function • Vertex (0,0) • f(x)=x2 • f(x)=3x2 • f(x)=½x2 • f(x)=-4x2 • f(x)=-¾x2

  5. The Quadratic FunctionStandard form • Parameters • a – same as for the basic equation, causes a vertical scale change • h – horizontal translation • k – vertical translation

  6. The Quadratic FunctionStandard form • Vertex (h,k) • h – the x-coordinate • k – the y-coordinate • f(x)=(x-2)2+5 • h is 2 • K is 5 • Vertex (2,5)

  7. The Quadratic FunctionStandard form • f(x)=(x-1)2+1 • f(x)=(x-5)2-5 • f(x)=(x+4)2+3 • f(x)=(x+6)2-7

  8. The Quadratic FunctionGeneral form • Parameter a • Vertical stretch (same conditions as for basic function) • Parameter b • A vertical and horizontal shift of the vertex • b0 shift to the left • b0 shift to the right • Parameter c • Vertical translation of the graph. • c corresponds to the y-intercept of the function

  9. The Quadratic FunctionGeneral form • f(x)= x2 • f(x)= x2+2x • f(x)= x2-2x • f(x)= x2+10x • f(x)= x2-10x

  10. The Quadratic FunctionGeneral form • f(x)= x2+x • f(x)= x2+x+5 • f(x)= x2+x-5 • f(x)= x2+x+10 • f(x)= x2+x-10

  11. The Quadratic FunctionGeneral form • f(x)= 2x2+2x • f(x)= -x2+5x+5 • f(x)= -5x2-2x • f(x)= 2x2-5x-5 • f(x)= 3x2-x-10

  12. The Quadratic FunctionFactored form • Parameter a • Vertical stretch (same conditions as for basic function) • Parameter x1 and x2 • the two zeros • The values of x when f(x)=0 (or the x-intercepts of the graph)

  13. The Quadratic FunctionFactored form • f(x)=(x-1)(x-1) • f(x)=(x-1)(x+5) • f(x)=(x-5)(x-1) • f(x)=(x+5)(x-5)

  14. The Quadratic functionStandard form  General form • f(x)=4(x-4)2+64 4(x-4)(x-4)+64 4(x2-4x-4x+16)+64 4(x2-8x+16)+64 4x2-32x+64+64 4x2-32x+128 Like terms

  15. The Quadratic functionStandard form  Factored form • f(x)=4(x-4)2-64 4(x-4)2-64=0 4(x-4)2=64 (x-4)2= 16 x-4=±4 x=4±4 x1=4+4=8 x2=4-4=0 f(x)=4(x-8)(x) Add 64 +64 +64 Divide by 4 ÷4 ÷4 √ √ Take the square root Add 4 +4 +4

  16. The Quadratic functionFactored form  General form • f(x)=(x-3)(x-5) (x-3)(x-5) (x2-5x-3x+15) x2-8x+15 Like terms

  17. The Quadratic functionFactored form  Standard form • f(x)=(x-3)(x-5) • The two zeros are 3 and 5 • Solve for k by substituting h into the equation and solving for f(x)

  18. The Quadratic functionGeneral form  Factored/Standard form • Completing the square • f(x)= 3x2-12x+6 • 3x2-12x+6=0 • 3x2-12x=-6 • 3(x2-4x)=-6 • 3(x2-4x+4)=-6+12 • 3(x2-4x+4)=6 • 3(x-2)2=6 • f(x)=3(x-2)2-6 -6 (from each side) Factor the left side, GCF Complete the square (goal is to have a perfect square inside the brackets on the left side): currently you have the form ax2+b, you must add (b/2)2, so (4/2)2 =4 to the left side...add the same to the other side, but because everything inside the brackets is multiplied by 3 add 12 to the right side Factor the left side, perfect square Can keep in standard form or multiply out to get general form

  19. Properties of a Graph • Domain – all allowable values of x…read along the x-axis from left to right • Range – all allowable values of y…read along the y-axis from min (bottom) to max (top) • Variation • Increasing…read along the x-axis, this is when the graph is going up • Decreasing…read along the x-axis, this is when the graph is going down • Sign • Positive…read along the x-axis, this is when the graph is above the x-axis • Negative…read along the x-axis, this is when the graph is below the x-axis • Extrema • Minimum…the lowest point of the graph • Maximum…the highest point of the graph

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