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Quadratic Equations and Functions
Essential Question for Chapter 10: How can we model and evaluate real
world situations using quadratic functions?
Section 1: Exploring Quadratic Graphs
Objectives: To graph quadratic functions of the form y = ax2 & y = ax2 + c
1
Activity: Plotting Quadratic Curves
Graph the equations y = x2 and y = 3x2 on the same coordinate plane
2a Describe how the graphs are alike.
2b Describe how the graphs are different.
3
Predict how the graph of y = (1/3)x2 will be similar to and different
from the graph of y = x2
2
4
Graph y = (1/3)x2. Where your predictions correct? Explain.
Quadratic Function:
_____________________________________________________________
_____________________________________________________________
Standard form of a Quadratic Function:
_____________________________________________________________
_____________________________________________________________
Standard Form of a Quadratic Function
Quadratic Parent Function: _______________________________________
Parabola: _____________________________________________________
Real World Examples:
Axis of Symmetry:
_____________________________________________________________
Vertex: _______________________________________________________
3
Minimum
Maximum
If a > 0 in y = ax2 + bx + c
If a < 0 in y = ax2 + bx + c
The parabola opens upward
The parabola opens downward
Identify the Vertex
3)
1)
2)
4)
4
Graphing y = ax2
1) Make a table of values and graph the quadratic function y = ½x2.
x
y = ½x2
(x, y)
2) Make a table of values and graph the quadratic function y = -2x2.
x
y = ½x2
(x, y)
5
Comparing Widths of Parabolas on the Calculator
Put all 3 graphs into Y= in your calculator, then press Zoom Standard.
Graph the quadratic functions and order from widest to narrowest.
1) f(x) = - 4x2; f(x) = (1/4)x2, and f(x) = x2
2) y = x2; y = (1/2)x2, and y = -2x2
Note:
When m  n the graph of y = mx2 is wider than the graph of y = nx2
Graphing y = ax2 + c
1) How do the graphs below differ?
x
y = 2x2
y = 2x2 + 3
6
2) How do the graphs below differ?
x
y = x2
y = x2 – 4
3) Suppose you see an eagle flying over a canyon. The eagle is 30 feet
above the level of the canyon’s edge when it drops a stick from its
claws. The force of gravity causes the stick to fall toward Earth. The
function h = -16t2 + 30 gives the height of the stick h in feet after t
seconds. Graph this quadratic function.
x
h = -16t2 + 30
7
4) Suppose a squirrel is in a tree 24ft above the ground. She drops an
acorn. The function h = -16t2 + 24 gives the height of the acorn in
feet after t seconds. Graph this quadratic function.
x
h = -16t2 + 30
Homework: Practice 10-1
(Multiples of 3)
8
Section 2: Quadratic Functions
Objectives: To graph quadratic functions of the form y = ax2 + bx + c
y = 2x2 + 2x
y = 2x2 + 4x
Property: Graph of a Quadratic Function
y = 2x2 + 6x
9
Graphing y = ax2 + bx + c
1) Graph: y = -3x2 + 6x + 5
Step 1: Find the equation of the axis of symmetry and the
coordinates of the vertex.
a = -3 b = 6 c = 5
x
b
6
6


 1 The axis of symmetry is x = 1.
2a 2(3)  6
To find the y-coordinate substitute 1 in for x.
y = -3(1)2 + 6(1) + 5 = 8
The vertex is (1, 8)
Step 2: Find two other points on the graph.
Set up a table with x & y.

x
-1
0
1
2
3
Step 3: Graph
y
-4
5
8
5
-4





Set up a table with five ordered
pairs.
Put the vertex in the middle
Pick two points above and below
the vertex
Find your five ordered pairs
Notice how the points will reflex
each other
Graph
10
2) f(x) = x2 – 6x + 9
x
-1
0
1
2
3
y
-4
5
8
5
-4
11
3) In professional fireworks display, aerial fireworks carry “star”
upward, ignite them, and project them into the air.
Suppose a particular star is projected from an aerial firework at a
starting height of 520 feet with an initial upward velocity of 72 ft./s.
How long will it take the star to reach its maximum height? How far
above the group will it be?
The equation h = -16t2 + 72t + 520 gives the star’s height h in feet at
time t in seconds. Since the coefficient of t2 is negative, the curve
opens downward, and the vertex is the maximum point.
4) A ball is thrown into the air with an initial upward velocity of
48 ft./sec. Its height h in feet after t seconds is given by the function
h = -16t2 + 48t + 4.
a. In how many seconds will the ball reach its maximum height?
b. What is the ball’s maximum height?
Homework: Practice 10-2
Worksheet (Multiples of 3)
12
Section 3: Solving Quadratic Equations
Objectives: To solve quadratic equations by graphing and by using the
roots
Activity: Finding x-intercepts
1
Find the x-intercepts of each graph.
a
b
2a
Solve: 2x – 3 = 0
2b
Is the solution of 2x – 3 = 0 the same as the x-intercept of y = 2x – 3?
3
4a
Do the x-intercepts that you found in Question 1b satisfy the equation
x2 + 3x – 4 = 0?
Graph y = x2 + x – 6
4b
Find the x-intercepts of the graph of y = x2 + x – 6.
4c
Do the values you found in part (b) satisfy the equation x2 + x – 6?
13
Definition: Standard Form of a Quadratic Equation
Roots of the equation or Zeros of the Function:
_____________________________________________________________
_____________________________________________________________
Solving by Graphing
1) x2 – 4 = 0
y = x2 – 4
2) x2 = 0
y = x2
14
3) x2 + 4 = 0
4) x2 – 1 = 0
5) 2x2 + 4 = 0
y = x2 + 4
15
6) x2 – 16 = -16
Using Square Roots
1) 2x2 – 98 = 0
3) 3n2 + 12 = 12
2) t2 – 25 = 0
4) 2g2 + 32 = 0
5) A city is planning a circular duck pond for a new park. The depth of
the pond will be 4 ft and the volume will be 20,000ft3. Find the radius
of the pond to the nearest tenth of a foot. Use the equation
V = πr2h, where V is the volume, r is the radius, and h is the depth.
Calculator Activity
with Roots
Homework:
Page 567: # 1-21
16
Section 4: Factoring to Solve Quadratic Equations
Objectives: To solve quadratic equations by factoring
Zero-Product Property
Using the Zero-Product Property
1) (x + 5)(2x – 6) = 0
2) (x + 7)(x – 4) = 0
3) (3y – 5)(y – 2) = 0
4) (6k + 9)(4k – 11) = 0
5) (5h + 1)(h + 6) = 0
Solving by Factoring
1) x2 + 6x + 8 = 0
2) x2 – 8x – 48 = 0
17
3) 2x2 – 5x = 88
4) x2 – 12x = -36
5) The diagram shows a pattern for an open-top box. The total area of
the sheet of material used to manufacture the box is 288 in 2. The
height of the box is 3-in. Therefore, 3-in x 3-in squares are cut from
each corner. Find the dimensions of the box.
6) Suppose that a box has a vase with a width of x, a length of x + 1, and
a height of 2 in. It is cut from a rectangular sheet of material with an
area of 182 in2. Find the dimensions of the box.
Homework: Page 574:
# 1-25
18
Section 5: Completing the Square
Objectives: To solve quadratic equations by completing the square
*Completing the Square Activity*
Completing the Square:
_____________________________________________________________
_____________________________________________________________
Finding n to Complete the Square
1) x2 – 12x + n
2) x2 + 22x + n
Solving x2 + bx = c
1) x2 + 9x = 36
Solving x2 + bx + c = 0
3) x2 – 20x + 32 = 0
4) x2 + 5x + 30 = 0
5) x2 – 14x + 16 = 0
2) m2 – 6m = 247
19
6) Suppose a woodworker wants to build a tabletop like the one shown
at the right. If the surface area is 26ft2, what is the value of x?
7) 4a2 – 8a = 24
8) 5n2 – 3n – 15 = 10
Homework:
Page 582: # 1-25
20
Section 6: Using the Quadratic Formula
Objectives: To use the quadratic formula when solving equations and
choosing an appropriate method for solving a quadratic eqation.
Quadratic Formula:
_____________________________________________________________
_____________________________________________________________
Deriving the Quadratic Formula:
21
Quadratic Formula
*Quadratic Formula Song!*
Using the Quadratic Formula
1) x2 + 6 = 5x
2) x2 – 2x – 8 = 0
3) x2 – 4x = 117
4) 2x2 + 4x – 7 = 0
5) -3x2 + 5x – 2 = 0
6) 7x2 – 2x – 8 = 0
22
7) Suppose a football player kicks a ball and gives it an initial upward
velocity of 47 ft/s. The starting height of the football is 3ft. If no one
catches the football, how long will it be in the air?
(Hint: The vertical motion formula: h = -16t2 + vt + c)
8) A football player kicks a ball with an initial upward velocity of
38.4 ft/sec from a starting height of 3.5 ft.
a) Substitute the values into the vertical motion fomula. Let h = 0.
b) Sovle. If no one catches the ball, how long will it be in the air?
Round to the nearest tenth of a second.
23
Choosing an Appropriate Method
There are many methods for solving a quadratic equation. You can
always use the quadratic formula, but sometimes another method may be
easier.
Method
Graphing
Square Roots
Factoring
Completing the
Square
Quadratic
Formula
When to Use
Use if you have a graphing calculator handy
Use if the equation has not x term
Use if you can factor the equation easilty
Use if the x2 term is 1, but you cannot factor the equation
easily
Use if the equation cannot be factored easily or at all
1) 2x2 – 6 = 0
2) 6x2 + 13x – 17 = 0
3) x2 + 2x – 15 = 0
4) 16x2 – 96x + 45 = 0
5) x2 – 7x + 4 = 0
6) 13x2 – 5x + 21 = 0
7) x2 – x – 30 = 0
8) 144x2 = 25
Homework:
Page 588: # 1-17
24
Section 7: Using the Discriminant
Objective: To find the number of solutions of a quadratic equation
Discriminant:
_____________________________________________________________
_____________________________________________________________
 b  b 2  4ac
x
2a
Find the discriminant for each quadratic equation
y = x2 – 6x + 3
y = x2 – 6x + 9
y = x2 – 6x + 12
25
Discriminate is
negative.
Discriminate is zero.
Discriminate is
positive.
Property of Discriminant
For the quadratic equation ax2 + bx + c = 0, where a ≠ 0, you can use
the value of the discriminant to determine the number of solutions.
If b2 – 4ac > 0, there are two solutions
If b2 – 4ac = 0, there is one solution
If b2 – 4ac < 0, there are no solutions
Using the discriminant
Find the number of solutions
1) 3x2 – 5x = 1
2) x2 = 2x – 3
3) 3x2 – 4x = 7
26
4) 5x2 + 8 = 2x
5) A constuction worker on the ground tosses an apple to a fellow
worker who is 20 ft above the ground. The starting height of the
apple is 5ft. Its initial upward velocity is 30 ft/s. Will the apple reach
the second worker?
6) Suppose the same construction worker tosses an apple with an initial
upward velocity of 32 ft/s. Wil the apple reach the second worker.
Homework:
Page 594: # 1- 15 &
# 17