chapter 4: quadratic functions and factoring 4.1 graphing...
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4.1 Graphing Quadratic Functions in Standard forma quadratic function in standard form is written y = ax2 + bx + c, where a ≠ 0A quadratic Function creates a U-shaped graph called a parabola The vertex is the lowest/highest point of the parabola
Axis of symmetry is the line that divides the graph into two symmetric parts
The roots of the function (answers) are the two points that cross the x-axis (x-intercepts), when y = 0
VOCAB:
Chapter 4: Quadratic Functions and Factoring
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Practice Graphing Graph of a quadratic functionif a is positive, the parabola opens up
if a is negative, the parabola opens down
The vertex has an xcoordinate of
The axis of symmetry is written x =
b2a
b2a
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Practice Graphing Graph of a quadratic function
STEP 1: find the xcoordinate of the vertex by using
STEP 2: Make a table of values
STEP 3: Plot the points and create the parabola
b2avertex =
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Graph: y = x2 2x 3
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Graph: y = 2x2 x +2
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Graph: y = 2x2 7x8
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4.2 Graphing Quadratic Functions in Vertex or Intercept form
vertex form: standard form:
The graph of y = a(xh)2 + k is the parabola y = ax2 translated horizontally h units and vertically k units
Characteristics of the graph:• the vertex is (h,k)• The axis of symmetry is x = h• The graph opens up if a>0 and down if a<0
y
x
y = ax2
(0,0)
y = a(xh)2+k
(h,k)
hk
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Step 1: Identify h, a and k. Use a to determine if the parabola opens up or down
Graph a quadratic function in vertex form: y = a(x-h) 2+k
Step 2: Plot the vertex (h,k) and draw the axis of symmetry
Step 3: Evaluate the function for two values of x.
Step 4: Draw parabola through plotted points
STEP 1
STEP 2
STEP 3
STEP 4
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Example 2
Step 1: Identify h, a and k. Use a to determine if the parabola opens up or down
Graph a quadratic function in vertex form: y = a(x-h) 2+k
Step 2: Plot the vertex (h,k) and draw the axis of symmetry
Step 3: Evaluate the function for two values of x.
Step 4: Draw parabola through plotted points
STEP 1
STEP 2
STEP 3
STEP 4
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Characteristics of the graph y=a(xp)(xq):• The xintercepts are p and q• The axis of symmetry is halfway between (p,0) and (q,0) it has equation• The graph opens up if a>0 and down if a<0
y
x
y = ax2
(p,0) (q,0)
Intercept form: y=a(xp)(xq)
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Example 1
Step 1: Identify the x-intercepts
Graph a quadratic function in intercept form: y = a(x-p)(x-q)
Step 2: Find the coordinates of the vertex
Step 3: Draw parabolua through the vertex and x-intercepts
STEP 1
STEP 2
STEP 3
STEP 4
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Example 1
Step 1: Identify the x-intercepts
Graph a quadratic function in intercept form: y = a(x-p)(x-q)
Step 2: Find the coordinates of the vertex
Step 3: Draw parabolua through the vertex and x-intercepts
STEP 1
STEP 2
STEP 3
STEP 4
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Example: Change intercept form to standard form
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Example: Change intercept form to standard form
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Example: Change vertex form to standard form
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Example: Change vertex form to standard form
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4.3 Solve x2 + bx + c = 0 by factoring
ac
b
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8
6
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x2 5x + 6Factor
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x2 + 7x + 12
Factor
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x2 1x 6
Factor
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Solve
x2 + 3x 4 = 0
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Solvex2 + 5x 14 = 0
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Solvex2 + 12x = 27
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Example 1:
Special Factoring Patterns
Difference of Squares (DOS):
Perfect Square Trinomial:
a2 b2 = (ab)(a+b)
a2 + 2ab + b2 = (a+b)2
a2 2ab + b2 = (ab)2
4x2 9
Example 2:
Example 3:
x2 + 6x + 9
x2 8x + 16
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You try:
1. x2 + 16x + 64
2. 9x2 49
3. x2 18x + 81
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4.4 Solve ax2 + bx + c = 0 by factoring
2x2 3x 20Factor
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3x2 + 11x 4Factor
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7x2 31x + 12Factor
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6x2 11x + 3Factor
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5x2 8x + 3 = 0Solve
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3x2 + 14x + 15 = 0Solve
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7x2 + 11x 30 = 0Solve
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2x2 + 3x 5 = 0Solve
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Factor out monomials first
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4.5 Solve Quadratic Equations by Finding Square RootsSQUARE ROOT OF A NUMBERif b2 = a, then b is a square root of a
example: if 32 = 9, then 3 is a square root of 9.
VOCAB: positive square root: √ negative square root: -√ positive/negative square root: ±√ radicand: the number or expression inside a radical symbol.
MATH IS √ DUDE
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Simplifying radicalsProperties of Radicals
Product property √ab = √a√b
Quotient property √ = a
b√a
√b
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An expression with radicals is insimplest form if the following are true 1.) No perfect square factors other
than 1 are in the radicand
2.) No fractions are in the radicand
3.) No radicals appear in the denominator of a fraction
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Examples
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Examples
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Examples:rationalize denominators of fractions
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Examples: solve a quadratic
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Examples: solve a quadratic
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4.6 Perform Operations with Complex Numbers
Not all quadratic equations have real-number solutions. For example: x2 = -1 has no real-number solutions because the square of any real number x is never a negative number
To overcome this problem mathematicians created an expanded system of numbers using the imaginary unit i. Defined as not that i2 = -1. The imaginary unit can be used to write the square root of any negative number.
The square root of a negative numberProperty Example
1. If r is a positive real number, then
2. By Property (1), it follows that
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example:
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example:
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Complex Numbers:A complex number written in standard form is a number a+bi where a and b are real numbers. The number a is the real part of the complex number and the bi is imaginary part.
If b≠0 then a + bi is an imaginary number. If a = 0 and b ≠ 0, then a+bi is a pure imaginary number.
RealNumbers
ImaginaryNumbers
Pure ImaginaryNumbers
Complex Numbers: a + bi
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The sum and difference of complex numbersTo add (subtract) two complex numbers, add (subtract) their real parts and their imaginary parts separately.
1. Sum of complex numbers:
2. Difference of complex numbers:
examples:a:
b:c:
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examples:1:
2:
3:
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Multiply Complex Numbers:
a:
b:
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Multiply Complex Numbers:
1:
2:
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Complex Conjugates:Two complex numbers of the form a+bi and abi are called complex conjugates. The product of conjugates is always a real number.
Divide Complex Numbers: use complex conjugates
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Divide Complex Numbers: use complex conjugates
1:
2:
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Plot Complex NumbersPlot the complex numbers in the same complex plane.
a.
b.
c.
d.
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Find absolute values of complex numbersAbsolute Vale of a Complex Number:
a.
b.
The absolute value of a complex number z = a + bi, denoted
is a nonnegative real number defined as
This is the distance between z and the origin in the complex plane
examples:
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4.7 Complete the Squarex2 + bx +( )2 = (x + )2
b2
b2
To complete the square for the expression x2 + bx, add
Diagram:
x
x
b
xx2 bxx
x
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example 1: Solve a quadratic by finding square roots
step 1: Write left side as binomial squared
step 2: Take square root of each side.
step 3: Solve for x
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example 2: Make a perfect square trinomial
Step 1: Find half of the coefficient of x
Step 2: square the result of step 1
Step 3: Replace c with the result from step 2
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example 3: Solve x 2 + bx + c (a = 1)
step 1: write left side in the form x 2+bx
step 2: Complete the square and add it to both sides
step 3: Write left side as a binomial squared
step 4: Take square root of each side
step 5: Solve for x (simplify if necessary)
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example 4: Solve ax 2 + bx + c (a ≠ 1)
step 1: divide everything by a
step 2: write left side in the form x2 + bx
step 3: complete the square and add it to both sides
step 4: write the left side as a binomial squared
step 5: take square root of both sides
step 6: solve for x, simplify if necessary
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Extra ExamplesSolve the equation by completing the square
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Write a quadratic function in vertex form
step 1: complete the square and add it to both sides
step 2: write beginning part as binomial squared
step 3: solve for y
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Write a quadratic function in vertex form
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4.8 Use the quadratic formula and the discriminant
Quadratic Formula: The solutions of thequadratic equation ax2+bx+c=0 are
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Proving the Quadratic Formula!
ax2+bx+c = 0 where a≠0
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Example 1:x2 8x + 15 = 0
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Example 2:
2x2 + 6x + 2 = 1
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Example 3: One solution
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Example 4:One solution
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Example 5: Imaginary Solution
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Example 6: Imaginary Solution
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Word ProblemsObjects Dropped:
Objects Launched/Thrown:initial velocity
initial height
A juggler tosses a ball into the air. The ball leaves the juggler's hand 4 feet above the ground and has an initial vertical velocity of 40 feet per second. The juggler catches the ball when it falls back to a height of 3 feet. How long is the ball in the air?
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Word ProblemsObjects Dropped:
Objects Launched/Thrown:initial velocity
initial height
A basketball player passes the ball to a teammate. The ball leaves the player's hand 5 feet above the ground and has an initial vertical velocity of 55 feet per second. The teammate catches the ball when it returns to a height of 5 feet. How long is the ball in the air?
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Word ProblemsThe equation h = 16t2 + 20t + 6 gives the height, h, in feet of a basketball as a function of t, in seconds
a) What is the maximum height the ball reaches?
b) At what time does the ball hit the ground?
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Applications of the discriminantIn the quadratic formula the expression inside the radical is the discriminant
Discriminantb2-4ac
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Number of solutions of a quadraticConsider the quadratic equation ax2 + bx + c = 0:
• If b24ac is positive, then the equation has two solutions• if b24ac is zero, then the equation has one solution• if b24ac is negative, then the equation has no real solutions
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Example:a. x2 3x 4
b. x2 + 2x 1
c. 2x22x + 3
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Example:a. 3x2 + 5x 1
b. x2 + 10x 25
c. x2 2x + 4
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Match the graph with the discriminant:
a. b24ac = 2
b. b24ac = 0
c. b24ac = 3
c
A
b
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4.10 Write Quadratic Functions and Models
Example 1: Write a quadratic function in vertex form y = a(x-h)2+K
Vertex: (1, 2) Point: (3, 2)
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Example 2: Write a quadratic function in vertex form y = a(x-h)2+K
Vertex: (4, 5) Point: (2, 1)
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Example 3: Write a quadratic function in intercept form y = a(x-p)(x-q)
xintercepts: 2 and 5 Point: (6, 2)
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Example 4: Write a quadratic function in intercept form y = a(x-p)(x-q)
xintercepts: 1 and 4 Point: (3, 2)
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Example 5: Write a quadratic function in Standard form y = ax 2 + bx + c
points: (1, 3), (0, 4), (2, 6)
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Example 6: Write a quadratic function in Standard form y = ax 2 + bx + c
points: (1, 5), (0, 1), (2, 11)