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Quadratic Patterns and Transformations

Explore quadratic patterns and transformations with this interactive tool. Discover how changing the values of a and b affects the shape, vertex, line of symmetry, and maximum or minimum point of a parabola. Practice solving quadratic equations and identifying the y-intercept and x-intercepts.

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Quadratic Patterns and Transformations

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  1. 2nd Level Difference Test means quadratic x 0 1 2 3 4 5 6 y 0 1 4 9 16 25 36 x 0 1 2 3 4 5 6 y 0 3 12 27 48 75 108 x 0 1 2 3 4 5 6 y 0 1/2 2 9/2 8 25/2 18 +1 +1 +3 =3(1) +1/2 +2 +1 +3 +9 =3(3) +3/2 +2 +1 +5 +15 =3(5) +5/2 +2 +1 +7 +21 =3(7) +7/2 +2 +1 +9 +27 =3(9) +9/2 +2 +1 +11 +33 =3(11) +11/2

  2. We have discovered a pattern! Starting from the vertex, every parabola will have the y coordinates increase by the sequence of odd integers, 1, 3, 5, 7, 9, … for every time the x coordinate increases by 1. If a doesn’t equal 1, then we multiply the value of a to every odd number in the sequence. Vertex @ (0,0) 7 5 3 1

  3. 1, 3, 5, 7 x 3 3, 9, 15, 21 Vertex @ (0,0) 9 3

  4. 1, 3, 5, 7, 9 x 1/2 ½, 3/2, 5/2, 7/2, 9/2 9/2 7/2 Vertex @ (0,0) 5/2 3/2 1/2

  5. The vertex is located at ( h, k ) Notice sign change on the h! 2 6 The vertex is located at ( 1, -3 ) 1, 3, 5 x -2 -2, -6, -10 10 Negatives values move down.

  6. Some of the graphing problems in MyMathLab will require the transformation rules from Section 3.5. Click the parabola icon, click the origin, and this screen will appear. a: if negative, reflects over x-axis Vertical stretch or shrink b: if negative, reflects over y-axis Horizontal stretch or shrink bx – c: set equal to zero and solve for x. Horizontal Shift LT or RT. d: Vertical Shift UP or DOWN.

  7. Group the x terms with ( )’s and factor out a. Place a + ____ inside the ( )’ and – ____ outside. (+4)2 16 Fill the blanks with (b/2)2 (+8/2)2 I suggest not squaring the 4 inside the ( )’s because we will use the +4 again when we factor the trinomial.

  8. Group the x terms with ( )’s and factor out a. Place a + ____ inside the ( )’ and – ____ outside. However, this time the GCF of 3 goes in the blank outside the ( )’s! GCF of 3 needs to be multiplied to both values in the blanks. Fill the blanks with (b/2)2 (-4/2)2 (-2)2 3( ) 3( ) 4 Now write the ( Binomial )2 and simplify the outside.

  9. Vertex Formula Find the values of a and b. a b a = 3 and b = -6 Notice that a is the same in both formulas. Start to write your answer. Find the value of h. Find the value of k.

  10. Vertex Formula Find the values of a and b. a b a = 3 and b = -8 Find the value of h. Find the value of k.

  11. Find the Vertex, L.O.S., and Max or Min. L.O.S. = Line of Symmetry or Axis of Symmetry L.O.S. is always a vertical line that goes through the vertex. Vertical lines always have the equation x = h. Max or Min. Max or Min is always the k value of the vertex. If the a value is negative, then the vertex is a MAX. If the a value is positive, then the vertex is a MIN. Vertex = ( 1, -3 ) L.O.S.: x = 1 Max @ -3 Vertex = ( -5, 9 ) L.O.S.: x = -5 Min @ 9

  12. Write the equation of a parabola with the vertex at ( 4, -3 ) and contains the point ( 6, -11 ). ( h, k ) ( x, y ) We need the equation with the vertex. y = a(x – h)2 + k We are given the vertex. We are given a point on the graph. Solve the equation for a.

  13. Write the equation of a parabola with the points at (4, 4), (3, 10), and (7, 10). We need the equation of the Line of Symmetry and Vertex. Find the Midpoint of points (3, 10) & (7, 10) ( h, k )

  14. Compare the usefulness of each form of the quadratic equations The a value determines which way the parabola opens and the how the pattern 1, 3, 5, 7, … is affected. The c value is the y-intercept. (0, c) Find the Vertex ( h, k ). Find the Graph. Find the L.O.S. Find the Max or Min Find the x-intercepts. Solve for x. (___, 0)

  15. Find the following given the equation . -9 Find the y-intercept. ( 0, 4 ) Find the Vertex ( h, k ). V( 3, -5 ) Find the L.O.S. x = 3 Find the Max or Min Min @ -5 Find the x-intercepts.

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