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Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or maximum value. Solve problems involving a quadratic function’s minimum or maximum value.

Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

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Page 1: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

Lesson 2.2, page 273Quadratic Functions

ObjectivesRecognize characteristics of parabolas.

Graph parabolas.Determine a quadratic function’s minimum or maximum value.

Solve problems involving a quadratic function’s minimum or maximum value.

Page 2: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

Vocabulary

• Quadratic Function: a function that can be written in the form

f(x) = ax2 + bx + c, a ≠ 0. • Standard form: f(x) = a(x - h)2 + k• Parabola: graph of quadratic function; U-

shaped curve; symmetric• Symmetry: when folded, two sides match

exactly

Page 3: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

f(x) = a(x - h)2 + k

• Vertex: the point (h,k); highest or lowest point of parabola; on axis of symmetry

• a: describes steepness and direction of parabola• Axis of symmetry: x = h; fold line that divides parabola into

two matching halves• Minimum: if a > 0, parabola opens up; vertex is minimum

point or lowest point of parabola; k is the minimum value• Maximum: If a < 0, parabola opens down; vertex the

maximum point or lowest point of parabola; k is the maximum value

Page 4: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

Minimum (or maximum) function value for a quadratic occurs at the vertex.

• Identify the vertex of each graph . Find the minimum or maximum value.

• A. B.y

x

10

10

-10-10

y

x

10

10

-10-10

Page 5: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

ReviewSolving Quadratic Equations

Example: 3x2 – 2x = 21

Get in standard form. 3x2 – 2x – 21 = 0

Factor. (3x + 7)(x – 3) = 0

Set each factor equal to zero. 3x+7=0 or x–3 = 0

Solve. x = -7/3 or x = 3

Check.

Page 6: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

If no “B” term, can use square root property to solve for zeros.

For example, if x2 = 16, then x = ±4.

• Solve f(x) = 1 – (x – 3)2.

Page 7: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

When factoring or square root property won’t work, you can always use the QUADRATIC FORMULA

• Get in standard form: ax2 + bx + c = 0• Substitute for a, b, and c in formula.• Solve.• Check.

This works every time!!!

2 4

2

b b acx

a

Page 8: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

Try these. Solve using the Quadratic formula.

• 3x2 - 2x - 1 = 0 x2 + 2x = -3

Page 9: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

Graphing Quadratic Functions f(x) = a(x - h)2 + k

1. Determine whether the parabola opens up (a > 0) or down (a < 0).

2. Find the vertex, (h,k).3. Find x-intercepts by solving f(x) = 0.4. Find the y-intercept by computing f(0).5. Plot the intercepts, vertex, and additional

points. Connect with a smooth curve. Note: Use the axis of symmetry (x = h) to plot

additional points.

Page 10: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

Check Point 1, page 276• Graph f(x) = -(x – 1)2 + 4.

y

x10

10

-10

-10

Page 11: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

Check Point 2, page 276

• Graph f(x) = (x – 2)2 + 1.

y

x

10

10

-10-10

Page 12: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

f(x) = ax2 + bx + c

• If equation is not in standard form, you may have to complete the square to determine the point (h,k).

2( ) 2 4 3f x x x

Page 13: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

Graph of 2( ) 2 4 3f x x x

Page 14: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

Finding the vertex when in ax2 + bx + c form.

,2 2

b bf

a a

Vertex =

Page 15: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

Check Point 3, page 279

• Graph f(x) = -x2 + 4x +1. Identify domain & range. y

x10

10

-10

-10

Page 16: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

Re-CapQuadratic Equations & Functions

0a ax2 + bx + c = 0 or f(x) = ax2 + bx + c,

• “x” is squared. • Graph would be a curve.• Solutions are at the x-intercepts. The “zeros” are the

solutions or roots of the equation.• To solve: factor & use zero product property, take square

root of both sides (if no B term), or use other methods.

Page 17: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

See Example 4, page 279

Check Point 4, page 280: f(x) = 4x2 - 16x + 1000a) Determine, without graphing, whether the function has a

minimum value or a maximum value.

b) Find the minimum or maximum value and determine where it occurs.

c) Identify the function’s domain and range.

Page 18: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

Example 1- Application Problem

A baseball player swings and hits a pop fly straight up in the air to the catcher. The height of the baseball, in meters, t seconds after the hit is given by the quadratic function

How long does it take for the baseball to reach its maximum height? What is the maximum height obtained by the baseball?

13.349.4)( 2 ttth

Page 19: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

Solution

Vertex:

23.5 4.9 3.5 34.3(3.5) 1

234.3

61.025meters2( 4.9)

3.5seconds

bx h

a

Page 20: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

A farmer has 600 feet of fencing to enclose a rectangular field. If the field is next to a river, so no fencing is needed on one side, find the dimensions of the field that will maximize the area. Find the maximum area.

Example 2 - Application

Page 21: Lesson 2.2, page 273 Quadratic Functions Objectives Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or

Solution

x x

600 – 2x

Width (x) =

Length(600-2x) =

Max area =

Area = (600 2x)(x)

A(x) = 2x2 + 600x

Find Vertex – to find max.

= 150

y = 2(150)2 + 600(150)

= 45000

600

2( 2)x

150

300

45000