Quadratic Functions and Quadratic Functions and ModelsModels

Section 3.1Section 3.1

•ObjectivesObjectives• Learn basic concepts about quadratic Learn basic concepts about quadratic

functions and their graphsfunctions and their graphs• Apply the vertex formulaApply the vertex formula• Sketch a quadratic functionSketch a quadratic function• Solve applications and model dataSolve applications and model data• Use quadratic regression to model Use quadratic regression to model

data.data.

BasicsBasics• The quadratic is a nonlinear function.The quadratic is a nonlinear function.• Its standard form is Its standard form is

f(x) = axf(x) = ax22 + bx +c + bx +c• The vertex form is The vertex form is

f(x) = a(x – h)f(x) = a(x – h)22 + k + kwhere (h,k) is the vertexwhere (h,k) is the vertex

Identify the functionIdentify the function• Y = 5x – 1Y = 5x – 1

• Y = 2xY = 2x22 + 1 + 1

• Y = (3xY = (3x22 + 1) + 1)22

• y = 1/(xy = 1/(x22 – 4) – 4)

Graph of a Quadratic FunctionGraph of a Quadratic Function

Shape of GraphShape of Graph• Called a parabolaCalled a parabola• If If aa is +, then graph opens up is +, then graph opens up• If If aa is -, then graph opens down is -, then graph opens down• The larger |The larger |aa| is, the skinnier the | is, the skinnier the

graph.graph.• The smaller |The smaller |aa| is, the fatter the | is, the fatter the

graph.graph.

VertexVertex

A

Axis of SymmetryAxis of Symmetry

How to Find the VertexHow to Find the Vertex• Y = -3(x – 1)Y = -3(x – 1)22 + 2 + 2

• The x-coordinate is the 1 and the y-The x-coordinate is the 1 and the y-coordinate is the 2, so (1,2)coordinate is the 2, so (1,2)

• The negative is part of the vertex form The negative is part of the vertex form and not part of the vertex.and not part of the vertex.

• Y = 5(x + 2)Y = 5(x + 2)22 – 5 – 5• Rewrite in vertex form. Y = 5(x – (- 2))Rewrite in vertex form. Y = 5(x – (- 2))22

– 5– 5• So the vertex is (-2, -5).So the vertex is (-2, -5).

How to Find the Axis of SymmetryHow to Find the Axis of Symmetry

• The axis of symmetry is a vertical The axis of symmetry is a vertical line through the vertex. line through the vertex.

• Since it is a line, it has an equation.Since it is a line, it has an equation.• Since it is vertical, it is always in the Since it is vertical, it is always in the

form of x = h (the x-value of the form of x = h (the x-value of the vertex).vertex).

ExampleExample• The sign of the The sign of the

leading coefficient leading coefficient (a) is ?(a) is ?

• What is the vertex?What is the vertex?• What is the What is the

equation of the equation of the axis of symmetry?axis of symmetry?

How to Find the VertexHow to Find the Vertex• When quadratic function is in standard When quadratic function is in standard

form, use the vertex formula to find form, use the vertex formula to find the vertex.the vertex.

• Vertex formulaVertex formula• X-coordinate found by –b/2a, where a is X-coordinate found by –b/2a, where a is

the leading coefficient (xthe leading coefficient (x22 term) and b is term) and b is the coefficient of the x term.the coefficient of the x term.

• Y-coordinate found by plugging above Y-coordinate found by plugging above answer in standard form and solving for y.answer in standard form and solving for y.

How to Find the VertexHow to Find the Vertex• Y = 3xY = 3x22 – 4x + 1 – 4x + 1

• a = 3 and b = -4 and c = 1a = 3 and b = -4 and c = 1• X-coordinate = -b/2a = -(-4)/2(3) = 4/6 X-coordinate = -b/2a = -(-4)/2(3) = 4/6

=2/3=2/3• Y-coordinate = 3(2/3)Y-coordinate = 3(2/3)22 – 4(2/3) + 1 = - – 4(2/3) + 1 = -

1/31/3• So the vertex is (2/3, -1/3)So the vertex is (2/3, -1/3)

• Y-coordinate can also be found byY-coordinate can also be found by(4ac – b(4ac – b22)/(4a).)/(4a).

How to Write in Vertex FormHow to Write in Vertex Form• Identify the leading coefficient, a.Identify the leading coefficient, a.• Find the vertex.Find the vertex.• Put in the vertex form.Put in the vertex form.• ExampleExample

• Y = xY = x22 – 7x + 5 – 7x + 5• A = 1, Vertex = (3.5, -7.25)A = 1, Vertex = (3.5, -7.25)• So the vertex form of the function isSo the vertex form of the function is

y = 1(x – 3.5)y = 1(x – 3.5)22 – 7.25 – 7.25

How to Get Formula From How to Get Formula From GraphGraph

• Identify vertex Identify vertex (-2,-2) and plug into (-2,-2) and plug into vertex form.vertex form.

• Pick another obvious Pick another obvious point, and plug into x point, and plug into x and y in vertex form. and y in vertex form.

• Solve for a.Solve for a.• Then put vertex and Then put vertex and

a back in vertex a back in vertex form.form.

How to Get Formula From How to Get Formula From GraphGraph

• Vertex (-2,-2)Vertex (-2,-2)• (2, 8)(2, 8)• 8= a(2 – (-2))8= a(2 – (-2))22 – 2 – 2• 8 = a(4)8 = a(4)22 – 2 – 2• 8= 16a – 28= 16a – 2• 10 = 16a10 = 16a• 10/16 = a = 5/810/16 = a = 5/8• y= 5/8(x + 2)y= 5/8(x + 2)22 – 2 – 2

ORRRRRRORRRRRRYou can plug the vertex and two other You can plug the vertex and two other points into L1 and L2 in your calculator points into L1 and L2 in your calculator and do a QuadReg. This will give you a.and do a QuadReg. This will give you a.

Suppose the vertex is (-1, 3) and the other Suppose the vertex is (-1, 3) and the other points on the graph are (-2, 1) and (0,1).points on the graph are (-2, 1) and (0,1).

Stat/edit, under L1 put -1 and -2 and 0. Stat/edit, under L1 put -1 and -2 and 0. Under L2 put 3 and 1 and 1.Under L2 put 3 and 1 and 1.

Stat/calc, #5 QuadReg, get the a value.Stat/calc, #5 QuadReg, get the a value.

ExampleExample• Sign of the leading Sign of the leading

coefficient?coefficient?• Vertex?Vertex?• Axis of symmetry?Axis of symmetry?• Increasing/Increasing/

decreasing intervals?decreasing intervals?

#83 page 185#83 page 185• A farmer has 1000 feet of fence to enclose A farmer has 1000 feet of fence to enclose

a rectangular area. What dimensions for a rectangular area. What dimensions for the rectangle result in the maximum area the rectangle result in the maximum area enclosed by the fence?enclosed by the fence?

• Perimeter = 2L + 2WPerimeter = 2L + 2W• 1000 = 2L + 2W1000 = 2L + 2W• 500 = L + W500 = L + W• 500 – W = L500 – W = L

• Area = LWArea = LW• A = (500 – W)W A = (500 – W)W • A= 500W – WA= 500W – W22

#83 page 185#83 page 185• Area is a quadratic function with x values Area is a quadratic function with x values

of width of the enclosure and y values of of width of the enclosure and y values of the area corresponding to that width. The the area corresponding to that width. The graph of the function opens downward, so graph of the function opens downward, so it has a maximum value…at the vertex.it has a maximum value…at the vertex.

• Find the vertex of the parabola, the x Find the vertex of the parabola, the x value is the width and the y value is the value is the width and the y value is the area. area.

• Plug x into 500 = L + W to get the lengthPlug x into 500 = L + W to get the length• 250 feet x 250 feet250 feet x 250 feet