Thinking Mathematically

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Thinking Mathematically

• Quadratic Functions and Their Graphs

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The Vertex of a Parabola

• For a parabola whose equation is
• y ax2 bx c
• The x-coordinate of the vertex is -b/(2a).
• The y-coordinate of the vertex is found by
  substituting the x-coordinate into the parabolas
  equation and evaluating y.

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Graphing Quadratic Functions

• The graph of y ax2 bx c, called a parabola,
  can be graphed using the following steps.
• Determine whether the parabola opens upward or
  downward. If a gt 0, it opens upward. If a lt 0,
  it opens downward.
• Determine the vertex of the parabola. The
  x-coordinate is -b/(2a). The y-coordinate is
  found by substituting the x-coordinate in the
  parabolas equation.

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Graphing Quadratic Functions cont.

• Find any x-intercepts by replacing y with 0.
  Solve the resulting quadratic equation for x.
• Find the y-intercept by replacing x with 0.
• Plot the intercepts and the vertex.
• Connect these points with a smooth curve that is
  shaped like a cup.

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Example Graphing a Parabola

• Graph the quadratic function y x2 - 2x - 3.

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Solution

• y x2 - 2x - 3
• Step 1 Determine how the parabola opens. Note
  that a, the coefficient of x2, is 1. Thus, a gt 0
  this positive value tells us that the parabola
  opens upward.
• Step 2 Find the vertex. The x-coordinate of the
  vertex is -b/(2a) -(-2)/(2(1)) 2/2 1. The
  y-coordinate of the vertex is 12-21-3 1 - 2 -
  3 -4. The vertex is (1, -4).

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Solution cont.

• y x2 - 2x - 3
• Step 3 Find the x-intercepts. Replace y with 0
  to get 0 x2 - 2x - 3. We can solve this
  equation by factoring.
• x2 - 2x - 3 0
• (x - 3)(x 1) 0
• (x - 3) 0 or (x 1) 0
• x 3 x -1
• Step 4 Find the y-intercept. Replace x with 0
  to get y 02 - 20 - 3 0 - 0 - 3 -3
• The parabola passes through (3,0), (-1,0) and
  (0,-3).

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Solution cont.

• y x2 - 2x - 3
• Steps 5 and 6 Plot the intercepts and the vertex.
  Connect these points with a smooth curve.

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Thinking Mathematically

• Quadratic Functions and Their Graphs