Class Notes: Graphing Quadratic Functions in Intercept Form

Objective

Learn how to graph a quadratic function when given in intercept form:

y = a(x - p)(x - q)

• a: leading coefficient
• p, q: x-intercepts

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Key Concepts

1. Identifying Features

• Intercept Form:
  y = a(x - p)(x - q)
  • ( p ), ( q ): x-intercepts at points ((p, 0)) and ((q, 0))
  • If there is no number before each parentheses, assume coefficient 1.
• a > 0: Parabola opens upward (minimum point)
• a < 0: Parabola opens downward (maximum point)

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2. Steps to Graph

Step 1: Find x-intercepts

• Set ( y = 0 ); solve for ( x ):
  ( x = p ), ( x = q )
• Plot ((p, 0)) and ((q, 0))

Step 2: Find Vertex

• The x-coordinate of the vertex (( h )): ( h = \frac{p + q}{2} )
• The y-coordinate (( k )):
  Substitute ( x = h ) into the original equation to find ( k ).
• Vertex is at ((h, k))

Step 3: Axis of Symmetry (AOS)

• Line through vertex: ( x = h )

Step 4: Additional Points for Symmetry

• Choose values of ( x ) equidistant from vertex (e.g. one left, one right)
• Compute ( y ) for those values

Step 5: Plot Points and Draw Parabola

• Common to have 5 points: both x-intercepts, y-intercept (if easy), vertex, and one symmetric point on each side.
• Draw the parabola smoothly through these points.

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Example 1

Equation:
( y = (x - 1)(x - 3) )

• ( a = 1 ) (opens upward)
• ( p = 1 ) ⇒ ( (1, 0) )
• ( q = 3 ) ⇒ ( (3, 0) )

Vertex:

• ( h = \frac{1 + 3}{2} = 2 )
• ( k = (2 - 1)(2 - 3) = (1)(-1) = -1 )
• Vertex: ( (2, -1) )

Other Points:

• ( x = 0: y = (0 - 1)(0 - 3) = (-1)(-3) = 3 ) ⇒ ( (0, 3) )
• ( x = 4: y = (4 - 1)(4 - 3) = (3)(1) = 3 ) ⇒ ( (4, 3) )

Summary Table:

| x | y | |---|----| | 0 | 3 | | 1 | 0 | | 2 | -1 | | 3 | 0 | | 4 | 3 |

Axis of Symmetry: ( x = 2 )
Minimum Value: ( y = -1 ) at ( x = 2 )

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Example 2

Equation:
( y = -2(x - 1)(x + 3) )

• ( a = -2 ) (opens downward)
• ( p = 1 ) ⇒ ( (1, 0) )
• ( q = -3 ) ⇒ ( (-3, 0) )

Vertex:

• ( h = \frac{1 + (-3)}{2} = \frac{-2}{2} = -1 )
• Compute ( k ):
  • ( x = -1 )
  • ( y = -2((-1) - 1)((-1) + 3) = -2(-2)(2) = -2 \times -4 = 8 )
• Vertex: ( (-1, 8) )

Other Points:

• ( x = 0: y = -2(0 - 1)(0 + 3) = -2(-1)(3) = -2 \times -3 = 6 ) ⇒ ( (0, 6) )
• ( x = -2: y = -2((-2) - 1)((-2) + 3) = -2(-3)(1) = -2 \times -3 = 6 ) ⇒ ( (-2, 6) )

Summary Table:

| x | y | |----|----| | -3 | 0 | | -2 | 6 | | -1 | 8 | | 0 | 6 | | 1 | 0 |

Axis of Symmetry: ( x = -1 )
Maximum Value: ( y = 8 ) at ( x = -1 )

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General Tips

• Intercept form is especially useful for quickly finding zeros and graphing.
• The vertex is always halfway between the x-intercepts for symmetric parabolas.
• Remember, the sign and value of a determine the shape and direction of the parabola.
• Y-intercept: Substitute ( x = 0 ) in the original equation.

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Practice Problem

Try graphing ( y = 2(x - 2)(x + 4) ):

• Identify: ( p = 2 ), ( q = -4 ), ( a = 2 )
• Find x-intercepts, vertex, axis of symmetry, y-intercept, plot points, and sketch.

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Now you know how to graph a quadratic function in intercept form!