1.2 Graphing Quadratic Functions In Vertex or Intercept Form

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1.2 Graphing Quadratic Functions In Vertex or
Intercept Form

• Definitions
• 2 more forms for a quad. function
• Steps for graphing Vertex and Intercept
• Examples
• Changing between eqn. forms

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Vertex Form Equation

• ya(x-h)2k
• If a is positive, parabola opens up
• If a is negative, parabola opens down.
• The vertex is the point (h,k).
• The axis of symmetry is the vertical line xh.
• Dont forget about 1 point on either side of the
  vertex! (3 points total!)

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Example 2 Graphy-½(x3)24 ya(x-h)2k

• a is negative (a -½), so parabola opens down.
• Vertex is (h,k) or (-3,4)
• Axis of symmetry is the vertical line x -3
• Table of values x y
• -1 2
• -2 3.5
• -3 4
• -4 3.5
• -5 2

Vertex (-3,4)
(-4,3.5)
(-2,3.5)
(-5,2)
(-1,2)
x-3
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Now you try one!ya(x-h)2k

• y2(x-1)23
• Open up or down?
• Vertex?
• Axis of symmetry?
• Table of values with 3 points?

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(-1, 11)
(3,11)
X 1
(0,5)
(2,5)
(1,3)
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Civil Engineering
where x and y are measured in feet. What is the
distance d between the two towers ?
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SOLUTION
The vertex of the parabola is (1400, 27). So, a
cables lowest point is 1400 feet from the left
tower shown above. Because the heights of the two
towers are the same, the symmetry of the parabola
implies that the vertex is also 1400 feet from
the right tower. So, the distance between the two
towers is d 2 (1400) 2800 feet.
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Intercept Form Equation

• ya(x-p)(x-q)
• The x-intercepts are the points (p,0) and (q,0).
• The axis of symmetry is the vertical line x
• The x-coordinate of the vertex is
• To find the y-coordinate of the vertex, plug the
  x-coord. into the equation and solve for y.
• If a is positive, parabola opens up
• If a is negative, parabola opens down.

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Example 3 Graph y-(x2)(x-4)ya(x-p)(x-q)

• Since a is negative, parabola opens down.
• The x-intercepts are (-2,0) and (4,0)
• To find the x-coord. of the vertex, use
• To find the y-coord., plug 1 in for x.
• Vertex (1,9)

• The axis of symmetry is the vertical line x1
  (from the x-coord. of the vertex)

(1,9)
(-2,0)
(4,0)
x1
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Now you try one!ya(x-p)(x-q)

• y2(x-3)(x1)
• Open up or down?
• X-intercepts?
• Vertex?
• Axis of symmetry?

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x1
(-1,0)
(3,0)
(1,-8)
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Football
a. How far is the football kicked ?
b. What is the footballs maximum height ?
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SOLUTION
a. Rewrite the function as y 0.026(x 0)(x
46). Because p 0 and q 46, you know the x -
intercepts are 0 and 46. So, you can conclude
that the football is kicked a distance of 46
yards.
b. To find the footballs maximum height,
calculate the coordinates of the vertex.
 
 
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WHAT IF? In Example 4, what is the maximum
height of the football if the footballs path can
be modeled by the function y 0.025x(x 50)?

SOLUTION
a. Rewrite the function as y 0.025(x 0) (x
50). Because p 0 and q 50, you know the x -
intercepts are 0 and 50. So, you can conclude
that the football is kicked a distance of 50
yards.
b. To find the footballs maximum height,
calculate the coordinates of the vertex.
 
 
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The maximum height is the y-coordinate of the
vertex, or about 15.625 yards.
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Changing from vertex or intercepts form to
standard form

• The key is to FOIL! (first, outside, inside,
  last)
• Ex y-(x4)(x-9) Ex y3(x-1)28
• -(x2-9x4x-36) 3(x-1)(x-1)8
• -(x2-5x-36) 3(x2-x-x1)8
• y-x25x36 3(x2-2x1)8
• 3x2-6x38
• y3x2-6x11

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Change from intercept form to standard form
Write y 2 (x 5) (x 8) in standard form.
Write original function.
y 2 (x 5) (x 8)
Multiply using FOIL.
2 (x 2 8x 5x 40)
2 (x 2 3x 40)
Combine like terms.
2x 2 6x 80
Distributive property
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Change from vertex form to standard form
Write f (x) 4 (x 1)2 9 in standard form.
f (x) 4(x 1)2 9
Write original function.
4(x 1) (x 1) 9
Rewrite (x 1)2.
4(x 2 x x 1) 9
Multiply using FOIL.
4(x 2 2x 1) 9
Combine like terms.
Distributive property
4x 2 8x 4 9
4x 2 8x 13
Combine like terms.
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Assignment
p. 15, 3-39 every 3rd problem (3,6,9,12,)