Math 20 Pre-Calculus

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C. Quadratic Functions

• Math 20 Pre-Calculus
• P20.7
• Demonstrate understanding of quadratic functions
  of the form   yax²bxc and of their graphs,
  including vertex, domain and range, direction
  of, opening, axis of symmetry, x- and
  y-intercepts.

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Key Terms
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• Quadratic Functions occur in a wide variety of
  real world situations. In this unit we will
  investigate functions and use them in math
  modelling and problem solving.

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1. Vertex Form

• P20.7
• Demonstrate understanding of quadratic functions
  of the form   yax²bxc and of their graphs,
  including
• vertex
• domain and range
• direction of opening
• axis of symmetry
• x- and y-intercepts.

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1. Vertex Form

• Investigate p. 143

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• The graph of a Quadratic Function is a parabola
• When the graph opens up the vertex is the lowest
  point and when it opens down the vertex is the
  highest point

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• The y-coordinate of the vertex is called the min
  value or max value depending of which way it
  opens.
• The parabola is symmetrical about a line called
  the axis of symmetry. The line divides the graph
  into two equal halves, left and right.
• So if you know the a of s and a point you can
  find another point (unless the point is the
  vertex)

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• The A of S intersects the vertex
• The x-coordinate of the vertex is the equation of
  the A of S.

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• Quadratic Function in vertex form f(x)
  a(x-p)2q are very easy to graph.
• a, p, and q tell you what you need.
• (p,q) Vertex
• Opens up a
• Opens down a
• Larger a narrower parabola
• Smaller a wider parabola

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Example 1
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Example 2
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Example 3
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Example 4
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Key Ideas p.156
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Practice

• Ex. 3.1 (p.157) 1-14
• 4-18

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2. Standard Form

• P20.7
• Demonstrate understanding of quadratic functions
  of the form   yax²bxc and of their graphs,
  including
• vertex
• domain and range
• direction of opening
• axis of symmetry
• x- and y-intercepts.

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2. Standard Form
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• Recall that the Standard form of a quadratic
  function is
• f(x) ax2bxc or y
  ax2bxc
• Where a, b, c are real numbers and a ? 0
• a determines width of graph (smaller a wider
  graph) and opening (a up and a down)
• b shifts the graphs left and right
• c shifts the graph up and down

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• We can expand f(x) a(x-p)2q to get f(x)
  ax2bxc , which will allow us to see the
  relation between the variable coefficients in
  each.

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• So,
• b -2ap or
• And
• c ap2 q or q c ap2

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• Recall that to determine the x-coordinate of the
  vertex, you use x p.
• So the x-coordinate of the vertex is

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Example 1
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Example 2
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Example 3
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Key Ideas p.173
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Practice

• Ex. 3.2 (p.174) 1-9, 11-17 odds
• 5-25 odds

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3. Completing the Square

• P20.7
• Demonstrate understanding of quadratic functions
  of the form   yax²bxc and of their graphs,
  including
• vertex
• domain and range
• direction of opening
• axis of symmetry
• x- and y-intercepts.

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3. Completing the Square

• You can express a quadratic function in vertex
  form, f(x) a(x-p)2q or standard form f(x)
  ax2bxc
• We already know we can go from vertex to standard
  by just expanding
• However to graph by hand it is much easier if the
  function is in vertex form because we have the
  vertex, axis of symmetry and max or min of the
  graph

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• So to be able to turn a standard form function
  into vertex form would be advantageous.
• This process is called Completing the Square

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• What we want to be able to do is rewrite the
  trinomial as a binomial squared. (x5)(x5)
  (x5)2

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• Lets complete the square

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• If there is a coefficient in front of the x2 term
  we have to add a couple steps.
• Complete the Square

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Example 1
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Example 2
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Example 3
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Example 4
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Key Ideas p.192
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Practice

• Ex. 3.3 (p.192) 1-9, 10-18 evens
• 1-9 odds in each, 10-28 evens