4.1: Do Now4.1: Do Now

Time (hours) 0 2 3 5 7Height (millimeters) 0 12 16 32 42

3

15

3

26

John placed a container outside during a rainstorm. A gauge on the side of the container shows the height, in millimeters, of the water in the container. The table below lists the height of water the gauge showed along with the corresponding number of hours after the rainstorm started. What is the average rate of change, in millimeters per hour, of the height of water in the container from time 2 hours to 5 hours?

a. b. c. 6 d. 8

Algebra IIAlgebra II

4.1: Graphing Quadratic Equations,4.1: Graphing Quadratic Equations,

HW: 4.1: p.240-242 (8, 12, 14, 22, 36, 38, HW: 4.1: p.240-242 (8, 12, 14, 22, 36, 38,

44-46 all, 58)44-46 all, 58)

Quiz 4.1-4.2: TBDQuiz 4.1-4.2: TBD

4.1: Notes4.1: Notes

Quadratic Equation:Quadratic Equation: Standard Form: y = axStandard Form: y = ax22 + bx + c + bx + c Parabolic shapeParabolic shape Vertical line of symmetryVertical line of symmetry If a is positive parabola will go up, if a is If a is positive parabola will go up, if a is

negative parabola will go down.negative parabola will go down.

NotesNotes

Quadratic Equation:Quadratic Equation: Through the line of symmetry is the vertex Through the line of symmetry is the vertex

which is either the maximum or minimum which is either the maximum or minimum value for the parabola.value for the parabola.• Vertex is maximum if parabola opens down.Vertex is maximum if parabola opens down.• Vertex is minimum if parabola opens up.Vertex is minimum if parabola opens up.

Notes continuedNotes continued Quadratic Equation continued. y = axQuadratic Equation continued. y = ax22 + bx + c + bx + c

Equation for Equation for line of symmetryline of symmetry::

x-coordinate of vertex (find y-value by plugging in x-coordinate of vertex (find y-value by plugging in x-value and solving:x-value and solving:

Tell whether the function has a minimum Tell whether the function has a minimum value or maximum value. Then find the value or maximum value. Then find the

minimum or maximum value.minimum or maximum value.

1.) y = -6x1.) y = -6x22 – 1 – 1 2.) f(x) = 2x2.) f(x) = 2x22 + 8x + 7 + 8x + 7

Steps to graphing a quadratic equationSteps to graphing a quadratic equation

Steps: y = axSteps: y = ax22 + bx + c + bx + c1.) Find the vertex.1.) Find the vertex.

2.) Graph using a table of values with the 2.) Graph using a table of values with the vertex in the middle.vertex in the middle.

Graph y = -2xGraph y = -2x22. . XX

YY

Graph y = -xGraph y = -x22 + 2 + 2. . XX

YY

Graph y = -4xGraph y = -4x22 + 8x + 2 + 8x + 2. . XX

YY

Do Now: p.242 #57Do Now: p.242 #57

Do NowDo NowThe figure shows the graph of the profit function for a company. In the graph, y represents the profit, in thousands of dollars, that the company earns for selling x thousand items. Interpret the meaning of the two intercepts shown in the context of the problem.

Algebra IIAlgebra II4.2: Graph quadratic function in vertex or 4.2: Graph quadratic function in vertex or

intercept formintercept form

HW: 4.2: p.249-251 (4, 8, 10, 16, 18, 34, 52, 54)HW: 4.2: p.249-251 (4, 8, 10, 16, 18, 34, 52, 54)

4.2: Vertex Form4.2: Vertex Form

y = a(x – h)y = a(x – h)22 + k + k Vertex: (h, k)Vertex: (h, k) Graph using tableGraph using table

Graph y = 2(x – 4)Graph y = 2(x – 4)22 – 1 – 1. . XX

YY

Graph y = (x + 1)Graph y = (x + 1)22 + 3 + 3. . XX

YY

Graph y = 2(x + 1)Graph y = 2(x + 1)22 + 3 + 3. . XX

YY

4.2: Intercept form4.2: Intercept form Steps in graphing intercept formSteps in graphing intercept form

y = a(x – p)(x – q)y = a(x – p)(x – q) Plot the x-intercepts: points p and qPlot the x-intercepts: points p and q Find x-coordinate of vertex by averaging p Find x-coordinate of vertex by averaging p

and q:and q:

Find y-coordinate of vertex by plugging in Find y-coordinate of vertex by plugging in x-coordinate and solving. Plot the vertex.x-coordinate and solving. Plot the vertex.

Graph y = (x + 2)(x – 2)Graph y = (x + 2)(x – 2) . .

Graph y = 3(x + 3)(x + 5)Graph y = 3(x + 3)(x + 5) . .

p.247 example 4p.247 example 4

3 types of graphs3 types of graphs1.) Stand. form: 1.) Stand. form: y = axy = ax22 + bx + c + bx + c

1.) x-coordinate of vertex: 1.) x-coordinate of vertex: 2.) chart with vertex in the middle2.) chart with vertex in the middle

2.) Vertex form: y = a(x – h)2.) Vertex form: y = a(x – h)22 + k + k1.) vertex: (h, k)1.) vertex: (h, k)2.) chart with vertex in the middle2.) chart with vertex in the middle

3.) Intercept form: 3.) Intercept form: y = a(x – p)(x – q)y = a(x – p)(x – q)1.) x-intercepts: p and q1.) x-intercepts: p and q

2.) x-coordinate of vertex: average 2.) x-coordinate of vertex: average p and q. Find y by plugging in.p and q. Find y by plugging in.

1.) y = 3x1.) y = 3x22

2.) y = (x – 4)(x – 2)2.) y = (x – 4)(x – 2)3.) y = -2x3.) y = -2x22 + 5 + 54.) y = ½x4.) y = ½x22

5.) y = (x – 2)5.) y = (x – 2)2 2

6.) y = 3x6.) y = 3x22 + 6x – 4 + 6x – 47.) y = ½(x + 1)(x – 2) 7.) y = ½(x + 1)(x – 2) 8.) f(x) = -x8.) f(x) = -x22 - 2x - 1 - 2x - 1

Graph and Graph and determine determine the domain the domain and range.and range.