Download Lesson 6-1: Maxima and Minima of Quadratic Functions

1
Lesson 5-7: Quadratic Functions in Vertex Form
Objectives: To graph quadratic functions in vertex form
To write the equation of a parabola in vertex form
In yesterday’s parabola investigation, we explored the vertex
2
form of a quadratic function, f  x   a  x  h   k
What does a do?
What does h do?
What does k do?
Where’s the AOS?
Where’s the vertex?
Example: Find the vertex, AOS, 2 other points and graph
1
y
y   ( x  4)2  3
2
x
What is the domain?
What is the range?
2
To find the equation of a parabola in vertex form, you
need to know the vertex and one other point.
Example: Find the equation of the graph below.
(0, -3)
(-2, -5)
Example: The Golden Gate Bridge is a suspension bridge. Its
main cable is shaped like a parabola and supported by two tall
towers. Find an equation to model the parabola, if the vertex
of the cable is 220 feet above the water and the two supporting
towers are 4200 feet apart and 750 feet above the water.
3
Lesson 5-1 & 5-7: Quadratic Functions in Standard Form
Objectives: To graph quadratic functions in standard form
Today, we will look at the standard (or polynomial form) of
the quadratic function,
f ( x)  ax 2  bx  c
b
The Axis of Symmetry (AOS) is given by x  2a
The Vertex is found by solving for y when
x
b
2a
The y-intercept is at (0, c)
In general, you should find the axis of symmetry, vertex,
y-intercept, and 1 other point, to draw the graph.
2
Example: Graph f ( x)  x  8x  9
AoS:
y
Vertex:
x
y-intercept:
other point:
2
Example: Graph y = 2x – 4
4
y
AoS:
Vertex:
y-intercept:
x
other point:
Domain
Range
Example: Rewrite y = 5(x – 2)2 + 4 in polynomial form. Find
all necessary information and graph.
AoS:
y
Vertex:
y-intercept:
other point:
Domain
Range
x
5
Lesson 5-1: Maxima and Minima of Quadratic Function
Objectives: To identify maxima and minima of a quadratics
Warm-Up:
2
Graph the quadratic function, f ( x)  x  3x  1 by finding
a) axis of symmetry
b) vertex
c) y intercept
d) one other point
Graph the quadratic function, f ( x)    x  1  4 by finding
2
a) axis of symmetry
b) vertex
c) y intercept
d) one other point
If a > 0, the graph of a parabola opens up and has a minimum
value at the vertex.
If a < 0, the graph of a parabola opens down and has a maximum
value at the vertex.
6
2
Example: Determine if the function f ( x)  x  4 x  9
has a maximum or minimum value and find the maximum or
minimum. State the Domain and Range of the graph.
Example: Susan throws a ball upward from a starting height
of 4 feet and with an upward velocity of 32 feet/sec. The
height of the ball as a function of time can be modeled by
h(t )  16t 2  32t  4
a) verify that at time t = 0, the ball is at 4 feet.
b) at what time will the ball reach its maximum height
c) what will the maximum height be?
You can use your graphing calculator to find minima and
maxima of graphs. First graph the function so the vertex is
visible in the window. Then use CALC 3:minimum or
4:maximum.
7
Example: A farmer has 400 feet of fencing to build a
rectangular pen of maximum area for his pigs. What
dimensions will produce a pen of maximum area? What is the
maximum area?
Example: The farmer has decided to use the side of his barn
as one side of the pen. He still has 400 feet of fencing for the
remaining three sides of the pen. What dimensions will
produce a pen of maximum area? What is the maximum area?
8
Lesson 5-2: Solving Quadratic Equations by Graphing
Objectives: To use graphs to solve quadratic equations
When a quadratic function is set equal to zero, the resulting
equation, ax2 + bx + c = 0, is called a quadratic equation. The
solution(s) to the quadratic equation are the values of x that
make the equation true. One method for finding the solution
is to graph the related quadratic function and find where it
crosses the x-axis. These values are called zeros or roots.
2
Example: Graph the quadratic function, f ( x)  x  4 x  3 by
finding
a) axis of symmetry
b) vertex
c) y intercept
d) one other point
What are the zeros (or roots) of the graph?
2
Example: Solve x  4 x  3  0 by graphing.
9
You can solve quadratic equations on the graphing
calculator by using CALC 2:zero.
Example: Use you graphing calculator to find the solutions
a) x2 – 3x = 0
b) 8x – x2 = 16
c) x2 + 4 = 0
In general, a quadratic equation can have two real solutions,
one real solution or no real solutions.
y
y
x
y
x
Example: The sum of two real numbers is 5 and their product
is -14. Find the two numbers.
x
10
Example: A volcanic eruption blasts a boulder upward
with an initial velocity of 240 ft/sec. Its height as a function
2
of time can be described by h(t )  16t  240t . How long
will it take the boulder to return to the same elevation it started
from?
11
5-3: Solving Quadratic Equations by Factoring
Objective: To solve quadratic equations by factoring
To write quadratic functions in intercept form
Another method for solving a quadratic equation is factoring.
This method uses the Zero Product Property:
If a•b = 0, then either a = 0 or b = 0
2
Example: Solve by factoring, x  7 x  6
Step 1: write as ax2 + bx + c = 0
Step 2: factor left side
Step 3: use the Zero Product Property to solve for x
Example: Solve by factoring. Don’t forget the right side must
equal zero. Check your solution by graphing on the calculator.
a) x2 + 6x = – 9
b) 5x2 – 13x = – 6
12
2
Example: Which solution to x  6 x is correct? Why?
Solution A
x2  6x
x2 6 x

x
x
x6
Solution B
x2  6 x
x2  6 x  0
x  x  6  0
x  0, x  6
Intercept (Factored) Form
Another form of the quadratic function is f ( x)  a( x  p)( x  q)
where p and q are roots of the equation. (Remember, roots are
the same as zeros and x-intercepts).
You can use the graph of a parabola to find the roots, p & q.
To find the value of a, you will need to know one other point.
Example: Write an equation for the graph shown in intercept
form. Then rewrite the equation in
standard form.
How can you find the vertex of a
parabola if you know its roots?
13
Lesson 5-4 Complex Numbers, Day 1
Objective: To define imaginary and complex numbers
To add and subtract complex numbers
2
Warm-up: Find the solution to the equation x  1  0
Until now, we have said that the above equation has no
solution since no real number can be squared to get 1 .
French mathematician Rene Descartes proposed that a number
2
i be defined so that i  1 or i  1
i is called the imaginary unit. Numbers such as 5i or 7i are
called pure imaginary numbers. Numbers such as 3 + 4i or
1
7  i are called complex numbers.
2
Complex Numbers
Real
Rational
Imaginary
Irrational
14
Complex numbers are considered equal if and only if
their real parts are equal and their imaginary parts are equal.
Example: Find the values of m and n such that
8  15i  2m  3ni
 m  2n    2m  n  i  5  5i
To add and subtract complex numbers, you combine like terms.
That is, you combine the real parts and the imaginary parts.
Simplify:
(6 – 4i) + (1 +3i)
(3 – 2i) – (5 – 4i)
15
An important property of the number i is that its powers
follow a pattern:
i  1
i 2  1
i 3  i 2  i  i
i4  i2  i2  1
i5  i 4  i  i
i6 
Simplify:
9
125 x5
2i  7i
10  15
3i 4i 5i 2
i11
i 31
i 5 i17
16
Lesson 5-4 Complex Numbers, Day 2
Objective: To multiply and divide complex numbers
You can use FOIL to multiply complex numbers.
(3 – 5i)(4 + 6i)
Example: In an AC electrical circuit, the voltage E, current I,
and impedance Z are related by the equation E = IZ.
If the AC current running through a circuit is 1 + 3j and the
impedance is 7 – 5j, find the voltage. (Note: electrical
engineers use the letter j rather than i for the imaginary unit so
it isn’t confused with the current I)
17
Just like you cannot have a radical in the denominator,
complex numbers cannot have i in the denominator. If the
denominator is purely imaginary, multiply both numerator and
denominator by i.
5i
2i
If the denominator is a complex number, multiply the
numerator and denominator by the complex conjugate.
3i
2  4i
2  4i
1  3i
18
Lesson 5-5: Completing the Square
Objectives:
 To solve quadratic equations by the Square Root Property
 To solve quadratic equations by completing the square
Warm-Up: Solve by factoring: x2 + 10x + 25 = 49
Another method for solving quadratics comes from the Square
Root Property:
If x 2  n, then x =  n
Note: The symbol + means “plus or minus.” It tells us that
there are two vlaues for x.
2
Example: Solve using Square Root Property: x  10 x  25  49
2
Example: Solve using Square Root Property: x  6 x  9  32
19
The Square Root Property can only be used when the
left side is a perfect square. If the left side is not a perfect
square, we can use a process called completing the square.
2
Example: Find the value of c that will make x  12 x  c
a perfect square.
Step 1: Find ½ of 12
Step 2: Square result of step 1
Step 3: Add result to x2 + 12x
Step 4: Rewrite as a perfect square
2
Example: Solve x  12 x  32  0 by completing the square
Step 0: Rewrite so left side is x2 + bx
Step 1: Find ½ of b
Step 2: Square result of Step 1
Step 3: Add result to both sides
Step 4: Rewrite left side as perfect square
Step 5: Take square root of both sides, don’t forget the +
Step 6: Solve both equations
2
20
If the leading coefficient of x is not 1, you must first
divide by the leading coefficient before completing the square.
Example: Solve 2x2 – 5x + 3 = 0 by completing the square
and check the result with your graphing calculator.
Not all solutions of quadratic equations will be real numbers.
Example: Solve x2 + 4x + 11 = 0 by completing the square.
21
Lesson 5-6: The Quadratic Formula
Objectives: To solve equations using the quadratic formula
To use the discriminant to determine the number and types of
roots
What methods do you know so far to solve quadratic
equations? What are the advantages and disadvantages of
each?
The last method for solving quadratic equations is the
quadratic formula. It is the easiest way to get the exact roots
of most equations.
b
b2  4ac
x

2a
2a
Note that the first term
b
,
2a
is the axis of symmetry and the
second term tells you how far to the right or left of the AOS
the roots are located. The AOS is always halfway between the
two roots.
22
How was the Quadratic Formula discovered? It comes
from Completing the Square.
Example 1: Solve x2 – 12x = 28 using the QF. Check w/ GC.
Example 2: Solve 2x2 + 4x – 5 = 0 using the QF. Check w/ GC.
2
23
Example 3: Solve x + 22x + 121 = 0 using the QF.
Check w/ GC.
Example 4: Solve x2 – 4x = -13 using QF. Check w/ GC.
The expression b2 – 4ac is called the discriminant. The value
of the discriminant determines the number and type of roots:
If b2 – 4ac is a perfect square  2 rational roots
If b2 – 4ac > 0 (but not a perfect square)  2 irrational
roots
If b2 – 4ac = 0  1 real root (it’s called a double root)
If b2 – 4ac < 0  2 complex roots
Example: Describe the number and type of roots for the
equation 2x2 – 16x + 33 = 0
24
Lesson 5-8: Graphing and Solving Quadratic Inequalities
Objective: To graph and solve quadratic inequalities
Warm up: Graph the quadratic equation y = x2 – 6x – 7
AOS:
Vertex:
y-intercept:
x-intercepts:
To graph a quadratic inequality:
1. Graph the related equality using a dashed line for < or >
and a solid line for < or >
2. Then select a point inside the parabola to see if it is a
solution to the inequality.
3. If it is a solution, shade inside the parabola. If it is not,
shade the region outside the parabola.
Example: Graph y < x2 – 6x – 7
25
To solve a quadratic inequality by graphing
1. Graph the related quadratic function
2. Use the graph to determine which values of x satisfy the
inequality.
Example: Solve x2 – 6x – 7 < 0
2
Example: Solve 0  2 x  6 x  1 by graphing with the
calculator.
Example: The height of a trampolinist above the ground
2
during one bounce is given by h(t )  16t  42t  3.75 , where
h is measured in feet and t is in seconds. For how long is she
at least 25 feet off the ground?
26
To solve a quadratic inequality algebraically,
1. Solve the related quadratic equation algebraically
2. Plot the solutions on a number line
3. Test values within each interval to see if they satisfy the
inequality
Example: Solve x2 + x > 6 algebraically
Example: Solve x2 + 11x+ 30 < 0 algebraically
27
Word Problem Review
Lots of different types of problems can be modeled using
quadratic functions. The solutions usually require finding the
maximum/minimum values or the zeros of the parabola.
1. The path of a football thrown across the field can be
2
modeled by the equation y  .005x  x  5 , where x
represents the distance in feet that the ball travels horizontally
and y represents the height of the ball. Find the maximum
height of the ball’s flight path.
2. A rectangular turtle pen is 6 feet long by 4 feet wide. The
pen is enlarged by increasing the length and the width by the
same amount in order to double the area. What are the new
dimensions of the pen?
28
Review of Quadratic Graphs and Equations
We have learned three different forms for a quadratic
equation. You should be able to find the key information from
any of the three forms.
Standard
Vertex
Intercept
AoS
Vertex
yintercept
xintercepts
Domain
Range
Example: Find all important information about the graph of
f ( x)  x2  30 x  64 . Then rewrite the equation in both
Vertex and Intercept form.