6.6 Analyzing Graphs of Quadratic Functions

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6.6 Analyzing Graphs of Quadratic Functions

• Write a Quadratic Equation in Vertex form

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CCSS A.SSE.3

• CHOOSE and PRODUCE an equivalent form of an
  expression to REVEAL and EXPLAIN properties of
  the quantity represented by the expression.
• a. FACTOR a quadratic expression to reveal the
  zeros of the function it defines.
• b. COMPLETE THE SQUARE iin a quadratic expression
  to REVEAL the maximum or minimum value of the
  function it defines.

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CCSS F.IF.7

• GRAPH functions expressed symbolically and SHOW
  key features of the graph, by hand in simple
  cases and using technology for more complicated
  cases.
• a. GRAPH linear and quadratic functions and show
  intercepts, maxima, and minima.

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Standards for Mathematical Practice

• 1. Make sense of problems and persevere in
  solving them.
• 2. Reason abstractly and quantitatively.
• 3. Construct viable arguments and critique the
  reasoning of others.  
• 4. Model with mathematics.
• 5. Use appropriate tools strategically.
• 6. Attend to precision.
• 7. Look for and make use of structure.
• 8. Look for and express regularity in repeated
  reasoning.

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Essential Questions

• How do I determine the domain, range, maximum,
  minimum, roots, and y-intercept of a quadratic
  function from its graph?
• How do I use quadratic functions to model data? 
• How do I solve a quadratic equation with non -
  real roots?

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Vertex form of the Quadratic Equation

• So far the only way we seen the Quadratic
  Equation is ax2 bx c 0.
• This form works great for the Quadratic Equation.
  
• Vertex form works best for Graphing.
• We need to remember how to find the vertex. The x
  part of the vertex come from part of the
  quadratic equation.

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Vertex form of the Quadratic Equation

• The x part of the vertex come from part of the
  quadratic equation.
• To find the y part, we put the x part of the
  vertex.
• The vertex as not (x, y), but (h, k)

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Find the vertex of the Quadratic Equation

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Find the vertex of the Quadratic Equation

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The Vertex form of the Quadratic Equation

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The Vertex form of the Quadratic Equation

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The Vertex form of the Quadratic Equation

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Write the Quadratic Equation in Vertex form

• Find a, h and k
• a 1
• h -1
• k 3

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Write the Quadratic Equation in Vertex form

• Find a, h and k
• a 1
• h -1
• k 3

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Vertex is better to use in graphing

• y 2(x - 3)2 2 Vertex (3 , -2)
• Put in 4 for x, y 2(3 - 4)2 2 (4, 0)
• Then (2, 0)
• is also a point

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Let see what changes happen when you change a
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Let see what changes happen when you change a
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Let see what changes happen when you change a
The larger the a, the skinner the graph What if
a is a fraction?
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Let see what changes happen when you change a

What if a is a fraction?
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What if we change h in the Vertex

• Let a 1, k 0

Changing the h moves the graph Left or Right.
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What if we change k in the Vertex

• Let a 1, h 0

k moves the graph up or down.
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Write an equation

• Given the vertex and a point on the graph.
• The vertex gives you h and k. We have to
  solve for a
• Given vertex (1, 2) and point on the graph
  passing through (3, 4)
• h 1 k 2

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Write an equation

• Given vertex (1, 2) and point on the graph
  passing through (3, 4)
• x3, y4

Solve for a
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Write an equation

• a ½

Solve for a
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Write an equation

• a ½

Final Answer
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6.7 Graphing and Solving Quadratic Inequalities
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Solving by Graphing

• Find the Vertex and the zeros of the Quadratic
  Equation
• You can find the zero in anyway we used in this
  chapter. Making Table
• Factoring
• Completing the Square
• Quadratic Formula

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Solving by Graphing

• Given Zeros

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Solving by Graphing

• Given Zeros

y x 2 -3x 2
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Solving by Graphing

• Which way do I shade?, inside or outside

y x 2 -3x 2
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Solving by Graphing

• The answers are in the shaded area

y gt x2 -3x 2
Why is it a dotted line?
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Solve x2 - 4x 3 gt 0

• Find the zeros, (x - 3)(x 1) 0
• x 3 x 1
• Is the Graph up or Down?

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Solve x2 - 4x 3 gt 0

• Find the zeros, (x - 3)(x 1) 0
• x 3 x 1
• Where do we shade? Inside or Outside

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Try a few points

• One lower then the lowest zero, one higher then
  the highest zero and one in the middle.
• Let x 0
• Let x 4
• Let x 2

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Try a few points

• One lower then the lowest zero, one higher then
  the highest zero and one in the middle.
• Let x 0 02 - 4(0) 3 gt 0 True
• Let x 4 42 4(4) 3 gt 0 True
• Let x 2 22 4(2) 3 gt 0 False
• Only shade where it is true.

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Solve x2 - 4x 3 gt 0
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Solve Quadratic Inequalities Algebraically

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Break the number line into three parts

• Test a number less then -2 x -2
• Let x - 3
• (-3)2 ( -3) 2 9 3 6
• 6 is not less then 2 False

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Break the number line into three parts

• Test a number between -2 and 1 -2 x 1
• Let x 0
• (0)2 (0) 2 0 0 0
• 0 is less then 2 True

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Break the number line into three parts

• Test a number great then 1 x 1
• Let x 2
• (2)2 (2) 2 4 2 6
• 6 is not less then 2 False

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Break the number line into three parts

• So the answer is -2 x 1