Pre-Calculus Honors Day14

2.1 Quadratic Functions- How do you write quadratic functions in standard form?- How to use quadratic functions to model and solve real-life problems?

The simplest type of quadratic function is f(x) = x2.

Quadratic Functions!Parabolas!

Opens Upward Opens Downward

Axis of Symmetry

Axis of Symmetry

Vertex: Minimum

Vertex: Maximum

• Quadratic Function:• f(x) = ax2 + bx +c; a ≠ 0

• Characteristics of Parabolas• Axis of symmetry: • Vertex: substitute x value from axis of symmetry to find the y

value of the vertex. (x, y)• If a > 0 (a = positive), parabola opens upward• If a < 0 (a = negative), parabola opens downward• Y-intercept (0, c)• X-intercepts: 0, 1, or 2, roots of solutions

• If b2 - 4ac = 0; 1 root (vertex)• If b2 - 4ac > 0; 2 roots• If b2 - 4ac < 0; no root

a

bx

2

Example 1: Find i) direction of opening ii) axis of symmetry iii) vertex iv) x and y – intercepts

82)(. 2 xxxfa

a

bx

2

11

)1(2

2

x

98)1(2)1()1( 2 f

i) a = 1, a >0, Opens UP

iii) Vertex: (-1, -9)

iv) y-intercept (x = 0) x-intercept (y=0) solve!

88)0(20)0( 2 f

(0, -8)

24

2040

)2)(4(0

820 2

xx

xx

xx

xx

(-4, 0) (2, 0)

ii) Axis of Symmetry:

Example 1: Find i) direction of opening ii) axis of symmetry iii) vertex iv) x and y – intercepts

742)(. 2 xxxfb

a

bx

2

11

)2(2

4

x

57)1(4)1(2)1( 2 f

i) a = -2, a <0, Opens DOWN

ii) Axis of Symmetry:

iii) Vertex:(1, -5)

iv) y-intercept (x = 0) x-intercept (y=0) solve!

77)0(4)0(2)0( 2 f

(0, -7)

7420 2 xx

No Real Roots!No x-intercepts

The Standard Form of a Quadratic Function

• Axis of Symmetry: • Vertical Line x = h

• Vertex: Point (h, k)• a > 0: Parabola opens upward• a < 0: Parabola opens downward

khxaxf 2)(

Example 2: Find i) direction of opening ii) axis of symmetry iii) vertex

3)4()() 2 xxha 1)2(2

1)() 2 xxgb

i) a = 1, a > 0, Opens UP i) a = -1/2, a < 0, Opens DOWN

ii) x = -4 ii) x = 2

iii) (-4, -3) iii) (2, 1)

Applications: Many applications involve finding the maximum and minimum value of a quadratic function.

• 1. If a > 0, f has a minimum that occurs at

• 2. If a < 0, f has a maximum that occurs at

a

bx

2

a

bx

2

Example 4: The path of a baseball is given by the function f(x) = -0.0032x2 + x + 3, where f(x) is the height of the baseball (in feet) and x is the horizontal distance from home plate (in feet). What is the maximum height reached by the baseball?

Hint: Find the distance (x) first, then use that to find the height

Height: 81.125 feet

Example5: The percent of income that Americans give to charities is related to their household income. For families with an annual income of $100,000 or less, the percent is approximately P = 0.0014x2 – 0.1529x + 5.855, 5≤ x ≤ 100where P is the percent of annual income given, and x is annual income (in thousands of dollars). What income level corresponds to the minimum percent of charitable contributions?

Income Level: 54.6 = $54,600

Example 6 on your own!• A textile manufacturer has daily production costs of C =

10,000-110x+0.45x2 where C is the total cost in dollars and x is the number of units produced. How many units should be produced each day to yield a minimum cost?

Example 7: What is the largest rectangular area that can be enclosed with 400 feet of fencing? What are the dimensions of the rectangle?

l =100ftw = 100ftA = 10,000 feet squared

Example 8: A farmer with 4000 meters of fencing wants to enclose a rectangular plot that borders on a river. If the farmer doesn’t fence the side along the river, what is the biggest area that can be enclosed?

A = 2,000,000 meters squared

Tonight’s Homework

Pg 143

#1-8 all, 23-26, 35, 36, 72, 73