Quadratic Function

Transformations of a Quadratic Function
  • A Transformations 
    • Vertical Stretch
    • Vertical Compression
    • Vertical Reflection
  • B Transformations 
    • Horizontal Stretch
    • Horizontal Compression
    • Horizontal Reflection​
  • C Transformations
    • Shift Left
    • Shift Right

  • D Transformations
    • Shift Up
    • Shift Down
​
Quadratic Parent Function
Transformations of the Quadratic Function
Single Transformations of the Quadratic Function
Link to A Transformations
This type of transformation is a vertical transformation
  • A = 1
    • no vertical dilation has occurred
  • A = -1
    • ONLY vertical reflection has occurred
  • | A | < 1 vertical compression has occurred
    • -1 < A < 0
      • vertical reflection and vertical compression have occurred
    • 0 < A < 1
      • ONLY vertical compression has occurred
  • | A | > 1 vertical stretch has occurred
    • A < -1
      • vertical reflection and vertical stretch have occurred
    • A > 1
      • ONLY vertical stretch has occurred
Link to B Transformations
This type of transformation is a horizontal transformation
  • B = 1
    • no horizontal dilation has occurred
  • B = -1
    • ONLY horizontal reflection has occurred
    • (difficult to see on symmetric figure)
  • | B | < 1 
    • horizontal stretch has occurred
      • -1 < B < 0  horizontal reflection and horizontal stretch have occurred
      • 0 < B < 1 ONLY   horizontal stretch has occurred
  • | B | > 1
    • horizontal compression has occurred
      • B > 1 ONLY horizontal compression has occurred
      • B < -1 horizontal reflection and horizontal compression have occurred
Link to C Transformations
This type of transformation is a horizontal transformation
  • C = 0 no transformation has occurred
  • C < 0  horizontal shift to the right has occurred
  • C > 0  horizontal ​shift to the left has occurred
Link to D Transformations
This type of transformation is a vertical transformation​
  • D = 0 no vertical shift has occurred
  • D < 0 vertical shift down has occurred
  • D > 0 vertical shift up has occurred
Two Transformations of the Quadratic Function
Link to AB Transformations
This type of transformation is a DILATION
  • This is really a weird new version of a vertical dilation
    • new A = old A(B)^2
  • We can still see the impact of the two dilation factors independently, but the actual type and magnitude of the vertical (or horizontal dilation) depends on the size of A and B and the sign of A
  • A = 1
    • no vertical dilation has occurred
  • A = -1
    • ONLY vertical reflection has occurred
  • | A | < 1 vertical compression has occurred
    • -1 < A < 0
      • vertical reflection and vertical compression have occurred
    • 0 < A < 1
      • ONLY vertical compression has occurred
  • | A | > 1 vertical stretch has occurred
    • A < -1
      • vertical reflection and vertical stretch have occurred
    • A > 1
      • ONLY vertical stretch has occurred
  • B = 1
    • no horizontal dilation has occurred
  • B = -1
    • ONLY horizontal reflection has occurred
    • (difficult to see on symmetric figure)
  • | B | < 1 
    • horizontal stretch has occurred
      • -1 < B < 0  horizontal reflection and horizontal stretch have occurred
      • 0 < B < 1 ONLY   horizontal stretch has occurred
  • | B | > 1
    • horizontal compression has occurred
      • B > 1 ONLY horizontal compression has occurred
      • B < -1 horizontal reflection and horizontal compression have occurred
Link to AC Transformations
This type of transformation is a DILATION and a HORIZONTAL Shift
  • The vertex of the related parabola lies on the x axis
  • The vertex of the related parabola is also its ONLY x intercept
  • When A is a perfect square, then this transformation is a perfect square trinomial (PST)
  • When A is NOT a perfect square, then this transformation is a MULTIPLE of perfect square trinomial (PST)
  • A = 1
    • no vertical dilation has occurred
  • A = -1
    • ONLY vertical reflection has occurred
  • | A | < 1 vertical compression has occurred
    • -1 < A < 0
      • vertical reflection and vertical compression have occurred
    • 0 < A < 1
      • ONLY vertical compression has occurred
  • | A | > 1 vertical stretch has occurred
    • A < -1
      • vertical reflection and vertical stretch have occurred
    • A > 1
      • ONLY vertical stretch has occurred
  • C = 0 no transformation has occurred
  • C < 0  horizontal shift to the right has occurred
  • C > 0  horizontal ​shift to the left has occurred
Link to AD Transformations
This type of transformation is a DILATION and a VERTICAL Shift
  • The vertex of the related parabola lies on the y axis
  • The vertex of the related parabola is the y intercept
  • When A is a perfect square, and the opposite of C is a perfect square then this transformation is a Difference of Two Squares Binomial (DOTS)
  • The parabola has either real conjugate roots or complex conjugate roots
  • A = 1
    • no vertical dilation has occurred
  • A = -1
    • ONLY vertical reflection has occurred
  • | A | < 1 vertical compression has occurred
    • -1 < A < 0
      • vertical reflection and vertical compression have occurred
    • 0 < A < 1
      • ONLY vertical compression has occurred
  • | A | > 1 vertical stretch has occurred
    • A < -1
      • vertical reflection and vertical stretch have occurred
    • A > 1
      • ONLY vertical stretch has occurred
  • D = 0 no vertical shift has occurred
  • D < 0 vertical shift down has occurred
  • D > 0 vertical shift up has occurred
Link to BC Transformations
This type of transformation is a DILATION and a HORIZONTAL Shift
  • B = 1
    • no horizontal dilation has occurred
  • B = -1
    • ONLY horizontal reflection has occurred (difficult to see on symmetric figure)
  • | B | < 1 
    • horizontal stretch has occurred
      • -1 < B < 0  horizontal reflection and horizontal stretch have occurred
      • 0 < B < 1 ONLY   horizontal stretch has occurred
  • | B | > 1
    • horizontal compression has occurred
      • B > 1 ONLY horizontal compression has occurred
      • B < -1 horizontal reflection and horizontal compression have occurred
  • -C/B = 0 no transformation has occurred
  • -C/B < 0  horizontal shift to the right has occurred
  • -C/B > 0  horizontal ​shift to the left has occurred
Link to BD Transformations
This type of transformation is a DILATION and a VERTICAL Shift
  • B = 1
    • no horizontal dilation has occurred
  • B = -1
    • ONLY horizontal reflection has occurred (difficult to see on symmetric figure)
  • | B | < 1 
    • horizontal stretch has occurred
      • -1 < B < 0  horizontal reflection and horizontal stretch have occurred
      • 0 < B < 1 ONLY   horizontal stretch has occurred
  • | B | > 1
    • horizontal compression has occurred
      • B > 1 ONLY horizontal compression has occurred
      • B < -1 horizontal reflection and horizontal compression have occurred
  • D = 0 no vertical shift has occurred
  • D < 0 vertical shift down has occurred
  • D > 0 vertical shift up has occurred
Link to CD Transformations
This type of transformation is a HORIZONTAL Shift and a VERTICAL Shift
  • C = 0 no transformation has occurred
  • C < 0  horizontal shift to the right has occurred
  • C > 0  horizontal ​shift to the left has occurred
  • D = 0 no vertical shift has occurred
  • D < 0 vertical shift down has occurred
  • D > 0 vertical shift up has occurred
Three Transformations of the Quadratic Function
Link to ABC Transformations
This type of transformation is a DILATION and a HORIZONTAL Shift
  • A = 1
    • no vertical dilation has occurred
  • A = -1
    • ONLY vertical reflection has occurred
  • | A | < 1 vertical compression has occurred
    • -1 < A < 0
      • vertical reflection and vertical compression have occurred
    • 0 < A < 1
      • ONLY vertical compression has occurred
  • | A | > 1 vertical stretch has occurred
    • A < -1
      • vertical reflection and vertical stretch have occurred
    • A > 1
      • ONLY vertical stretch has occurred
  • B = 1
    • no horizontal dilation has occurred
  • B = -1
    • ONLY horizontal reflection has occurred (difficult to see on symmetric figure)
  • | B | < 1 
    • horizontal stretch has occurred
      • -1 < B < 0  horizontal reflection and horizontal stretch have occurred
      • 0 < B < 1 ONLY   horizontal stretch has occurred
  • | B | > 1
    • horizontal compression has occurred
      • B > 1 ONLY horizontal compression has occurred
      • B < -1 horizontal reflection and horizontal compression have occurred
  • -C/B = 0 no transformation has occurred
  • -C/B < 0  horizontal shift to the right has occurred
  • -C/B > 0  horizontal ​shift to the left has occurred
Link to ABD Transformations
This type of transformation is a DILATION and a VERTICAL Shift
  • A = 1
    • no vertical dilation has occurred
  • A = -1
    • ONLY vertical reflection has occurred
  • | A | < 1 vertical compression has occurred
    • -1 < A < 0
      • vertical reflection and vertical compression have occurred
    • 0 < A < 1
      • ONLY vertical compression has occurred
  • | A | > 1 vertical stretch has occurred
    • A < -1
      • vertical reflection and vertical stretch have occurred
    • A > 1
      • ONLY vertical stretch has occurred
  • B = 1
    • no horizontal dilation has occurred
  • B = -1
    • ONLY horizontal reflection has occurred (difficult to see on symmetric figure)
  • | B | < 1 
    • horizontal stretch has occurred
      • -1 < B < 0  horizontal reflection and horizontal stretch have occurred
      • 0 < B < 1 ONLY   horizontal stretch has occurred
  • | B | > 1
    • horizontal compression has occurred
      • B > 1 ONLY horizontal compression has occurred
      • B < -1 horizontal reflection and horizontal compression have occurred
​
  • D = 0 no vertical shift has occurred
  • D < 0 vertical shift down has occurred
  • D > 0 vertical shift up has occurred
Link to BCD Transformations
This type of transformation is a DILATION, a HORIZONTAL, and a VERTICAL Shift
  • B = 1
    • no horizontal dilation has occurred
  • B = -1
    • ONLY horizontal reflection has occurred (difficult to see on symmetric figure)
  • | B | < 1 
    • horizontal stretch has occurred
      • -1 < B < 0  horizontal reflection and horizontal stretch have occurred
      • 0 < B < 1 ONLY   horizontal stretch has occurred
  • | B | > 1
    • horizontal compression has occurred
      • B > 1 ONLY horizontal compression has occurred
      • B < -1 horizontal reflection and horizontal compression have occurred
  • -C/B = 0 no transformation has occurred
  • -C/B < 0  horizontal shift to the right has occurred
  • -C/B > 0  horizontal ​shift to the left has occurred
  • D = 0 no vertical shift has occurred
  • D < 0 vertical shift down has occurred
  • D > 0 vertical shift up has occurred
Four Transformations of the Quadratic Function
Link to ABCD Transformations
This type of transformation is a DILATION, a HORIZONTAL, and a VERTICAL Shift
  • A = 1
    • no vertical dilation has occurred
  • A = -1
    • ONLY vertical reflection has occurred
  • | A | < 1 vertical compression has occurred
    • -1 < A < 0
      • vertical reflection and vertical compression have occurred
    • 0 < A < 1
      • ONLY vertical compression has occurred
  • | A | > 1 vertical stretch has occurred
    • A < -1
      • vertical reflection and vertical stretch have occurred
    • A > 1
      • ONLY vertical stretch has occurred

​
  • B = 1
    • no horizontal dilation has occurred
  • B = -1
    • ONLY horizontal reflection has occurred (difficult to see on symmetric figure)
  • | B | < 1 
    • horizontal stretch has occurred
      • -1 < B < 0  horizontal reflection and horizontal stretch have occurred
      • 0 < B < 1 ONLY   horizontal stretch has occurred
  • | B | > 1
    • horizontal compression has occurred
      • B > 1 ONLY horizontal compression has occurred
      • B < -1 horizontal reflection and horizontal compression have occurred
  • -C/B = 0 no transformation has occurred
  • -C/B < 0  horizontal shift to the right has occurred
  • -C/B > 0  horizontal ​shift to the left has occurred
  • D = 0 no vertical shift has occurred
  • D < 0 vertical shift down has occurred
  • D > 0 vertical shift up has occurred
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