Download 4.1 Graph Quadratic Functions in Standard Form (Parabolas)

4.1 Graph Quadratic Functions in Standard
Form (Parabolas)
Tuesday, June 08, 2010
12:31 PM
Axis of
symmetry
Vertex
Link to Parabola Animation
Another Cool Animation
Ch 4 Page 1
Graph:
Graph:
1.
2.
3.
4.
2. Graph:
3. Graph:
Graphing any quadratic function in standard form:
Find
To find the value of the vertex, plug in the value you found for the axis of symmetry into
the original equation and solve for .
Those 2 values are your vertex
- intercept:
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Graph:
Graph:
Maximum and Minimum values
• If the graph opens up then there is a minimum value
• If the graph opens down then there is a maximum value
• The max or min is the coordinate of the vertex.
Tell whether the function
Ch 4 Page 3
has a min or max value, then find it.
4.2 Graph Quadratic Functions (Parabolas) in
Vertex or Intercept Form
Wednesday, June 09, 2010
8:09 AM
Animated Parabola Movement
Vertex Form of a Parabola:
• The vertex is
-- *use the opposite of h
• the axis of symmetry is
•
opens up and
opens down
Graph a quadratic function in vertex form
Graph:
2. Graph:
1. Graph:
Ch 4 Page 4
1.
2.
3.
4.
5.
Intercept Form of a Parabola:
The -intercepts are and
*use opposite of and
The axis of symmetry is halfway between and
The -coordinate of the vertex:
The -coordinate is found by plugging in into the original
equation and solving for .
Graph:
Graph:
Changing an equation from intercept form to standard form
Write
in standard form. Multiply out the right and combine like terms.
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Changing an equation from vertex form to standard form.
Write
in standard form.
Write the quadratic function in standard form.
2.
1.
Graph:
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4.3 Solve x2 + bx + c = 0 by factoring
Wednesday, June 09, 2010
9:10 AM
Factor the expression.
1.
2.
3.
1.
2.
3.
Factoring a Difference of 2 Squares:
2.
1.
3.
Factoring Perfect Square Trinomials:
2.
1.
Solving quadratic equations
The solutions are called the roots of the equation.
To solve: 1. Set the equation = 0
2. Factor completely
3. Set your factors = 0 and solve for .
What are the roots of the equation:
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4.
and
3.
Find roots
Solve for
Find zeros
Find -int's
Solve the equation
2.
1.
3.
Zeros of a Function: In lesson 4.2 you learned that the -intercepts of the graph of
are and . Because the function's value is zero when
and
the numbers and are also called zeros of the function.
Find the zeros of the function by rewriting in intercept form.
2.
1.
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4.4 Solve
Wednesday, June 09, 2010
by Factoring
10:36 AM
Factor:
by grouping.
You try these
2.
1.
4.
3.
Factor Difference of squares and perfect square trinomials.
1.
2.
3.
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4.
Factoring out monomials(GCFs) first, then continue to factor completely.
2.
3.
1.
You try these
2.
1.
3.
Solve the quadratic equation by factoring and setting your factors = 0. Always start with GCF.
Finding the zeros of a function by rewriting it in intercept form.
Find the zeros of
Ch 4 Page 10
Quiz Review
Wednesday, June 09, 2010
11:22 AM
Graph:
Graph:
Graph:
Factor completely:
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Factor completely:
1.
2.
3.
4.
Solve each equation by factoring:
1.
2.
3.
4.
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4.5 Solve Quadratic Equations by
Finding Square Roots
Wednesday, June 09, 2010
11:24 AM
The expression
is called a radical. The symbol
is a radical sign, and the number
beneath the radical sign is the radicand of the expression.
Simplifying Square Roots
• No perfect square can be left in the radical
• No radicals can be left in the denominator
Simplify the expression.
Practice for Example 1
6.
4.
7.
2.
5.
3.
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8.
The expressions
and
are conjugates of each other. Their product is
always a rational number
Rationalizing the denominator - eliminating radicals from the denominator when simplifying.
Simplify.
1.
3.
2.
Simplify the expression.
4.
1.
5.
2.
6.
3.
Solving a quadratic equation
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Solving a quadratic equation
2.
1.
Solve the equation.
1.
2.
3.
Solve for x:
Simplify:
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4.6 Complex(Imaginary) Numbers
Thursday, June 10, 2010
9:33 AM
Cool Fractals are made with
Complex Numbers
Not all quadratic equations have real-number solutions. For example,
has no realnumber solutions because the square of any real number is never a negative number. To
overcome this problem, mathematicians created an expanded system of numbers using the
imaginary unit defined as
Note that
.
Here are some examples
Solve a quadratic equation.
1.
2.
1.
2.
3.
Adding and subtracting complex numbers - same as regular variables except you must
write the imaginary number last.
1.
2.
3.
Multiplying complex numbers - the same as regular variables except
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Multiplying complex numbers - the same as regular variables except
You cannot leave an in your final answer!
1.
2.
Dividing complex numbers - cannot leave an imaginary # in the denominator - so you must
multiply by conjugate
Write the quotient in standard form:
Write the expression as a complex number in standard form.
1.
2.
3.
4.
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4.7 Completing the Square
Thursday, June 10, 2010
11:25 AM
Solving quadratics by finding square roots.
1.
Solve
2.
when a =1 by completing the square.
1.
Solving
2.
-- when
Solve:
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-- by completing the square
*must divide all terms by coefficient of x2
1.
2.
Vertex Form:
where
is the vertex of the graph. To write a
quadratic function in vertex form, use completing the square.
Write:
in vertex form, then identify the vertex.
1.
2.
Find the maximum value of a quadratic function.
Baseball: The height (in feet) of a baseball seconds after it is hit is given by this
function:
Find the maximum height of the baseball.
Find the vertex.
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4.8 Using Quadratic Formula to Solve
Quadratic Equations
Thursday, June 10, 2010
12:35 PM
In order to use the quadratic formula, you
must first write the equation in standard form
Solve using the quadratic formula.
1.
2.
3.
4.
Ch 4 Page 20
Review for test
Monday, November 01, 2010
9:15 AM
Solve by completing the square.
1.
Simplify:
2.
3.
4. Simplify:
5. Solve for x.
6. Solve using the quadratic formula.
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7. Solve by factoring:
8. Graph:
Change into vertex form by completing the square and find the vertex.
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