Lesson Notes: Intercept Form of a Quadratic Equation

1. Quadratic Equation Forms

• Standard Form:
  ( y = ax^2 + bx + c )

  • Easier to find the y-intercept (at ( x = 0 ), ( y = c ))

• Intercept Form:
  ( y = a(x - p)(x - q) )

  • Easier to find the x-intercepts (( x = p ), ( x = q ))
  • ( p ), ( q ): x-intercepts (roots)
  • ( c ): y-intercept

• Vertex Form:
  ( y = a(x - h)^2 + k )

  • ( (h, k) ): Vertex of the parabola

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2. Finding Intercepts

• Y-intercept:

  • Set ( x = 0 ) in standard form:
    ( y = c )
  • Example: ( c = 10 ) ⇒ y-intercept at ( (0, 10) )

• X-intercepts:

  • Set ( y = 0 ), solve for ( x )
  • In intercept form, roots are ( x = p ) and ( x = q )
  • Example in intercept form: ( y = a(x - 1)(x - 5) )
    • x-intercepts: ( (1, 0) ), ( (5, 0) )

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3. Factoring Example (Standard to Intercept Form)

Given equation in standard form: ( y = 2x^2 - 12x + 10 )

• Factor out GCF ((2)):

  [ y = 2(x^2 - 6x + 5) ]

• Factor trinomial:

  • Find two numbers that multiply to (5), add to (-6): (-1) and (-5)
  • Factored form:
    ( y = 2(x - 1)(x - 5) )

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4. Finding the Vertex

• The x-coordinate (h) of the vertex is the average of the roots (( p ) and ( q )):
  [ h = \frac{p + q}{2} ] Example: ( (1 + 5) / 2 = 3 )

• Substitute ( x = 3 ) into the intercept or standard form to find ( k ): [ y = 2(3 - 1)(3 - 5) = 2(2)(-2) = -8 ] So, vertex at ( (3, -8) )

• Vertex Form Equation:
  [ y = 2(x - 3)^2 - 8 ]

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5. Axis of Symmetry (AOS)

• The AOS is the vertical line through the vertex: [ x = h ] In this example: ( x = 3 )

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6. Sum and Product of the Roots

Sum of Roots

• Intercept Form:
  Sum ( = p + q )

• Standard Form:
  Sum ( = -\frac{b}{a} )

• Vertex Form:
  Sum ( = 2h )

Product of Roots

• Intercept Form:
  Product ( = pq )

• Standard Form:
  Product ( = \frac{c}{a} )

• Vertex Form:
  Product ( = h^2 + \frac{k}{a} )

Example Work:

• ( p = 1, q = 5 ):
  Sum ( = 6 ), Product ( = 5 )
• For ( y = 2x^2 - 12x + 10 ):
  Sum ( = -\frac{-12}{2} = 6 ), Product ( = \frac{10}{2} = 5 )
• Vertex (( h = 3, k = -8, a = 2 )):
  Sum ( = 2 \times 3 = 6 ),
  Product ( = 3^2 + \frac{-8}{2} = 9 - 4 = 5 )

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7. Additional Resources

• Reference formula sheet and additional practice problems in the video description.

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Key Takeaways

• Choose the right form to make finding desired features easier:
  • Standard: y-intercept
  • Intercept: x-intercepts
  • Vertex: Vertex and axis of symmetry
• Know the formulas for sum and product of roots in all three forms.
• Factorization helps to convert between forms.