1. Grade 10 Academic (MPM2D) Unit 4: Quadratic Relations The Quadratic Relations (Vertex Form) – Transformations
Mr. Choi
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2. Quadratic Relations
The expression defines a quadratic relation in vertex form.
The coordinates of the vertex of the corresponding parabola are (p, q) .
If a > 0, the parabola opens upward. If a < 0, the parabola opens downward.
A quadratic relation in vertex form can be converted to standard form by expanding and collecting like terms.
A quadratic relation in standard form can be converted to vertex form by completing the squares which will be discussed in this unit.
Quadratic Relations (Vertex Form): Transformations
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3. Quadratic Relations (Vertex Form)
The expression defines a quadratic relation called the vertex form with a horizontal translation of p units and vertical translation of q units.
x – intercepts
y – intercepts
Quadratic Relations (Vertex Form): Transformations
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4. Summary (Translations)
Quadratic Relations (Translations)
y = (x - p)2
Descriptions / Mapping Rules
Examples
p > 0
p < 0
Translates p units Right
(x, y)  (x + p, y)
Translates p units Left
(x, y)  (x - p, y)
y = x2 + q
Descriptions / Mapping Rules
Examples
q > 0
q < 0
Translates q units Up
(x, y)  (x, y + q)
Translates q units Down
(x, y)  (x, y - q)
Quadratic Relations (Vertex Form): Transformations
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5. Quadratic Relations (Vertical Stretches)
Summary (Opening and Stretches)
Quadratic Relations (Vertical Stretches)
y = ax2
Descriptions / Mapping Rules
Examples
a > 1
0 < a < 1
Vertically expand by a factor of a
(x, y)  (x, ay)
Vertically compress by a factor of a
(x, y)  (x, ay)
Quadratic Relations (Opening)
y = ax2
Descriptions / Mapping Rules
Examples
a > 0
a < 0
Opens (Concaves) up
(x, y)  (x, +ay)
Opens (Concaves) down
(x, y)  (x, -ay)
Quadratic Relations (Vertex Form): Transformations
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6. Example 1: Parabola with Transformations
Find the vertex, the axis of symmetry, the direction of opening, x-intercept(s) and the y-intercept for the graph of the quadratic relation. State the mapping rule.
Vertex:
Vertical compresses by factor of 1/2.
Opens up
Right by 5
Down by 2
Mapping Rule:
(x , y)  (x + 5, ½y - 2) x y
x + 5
0.5y - 2 -2 4 -1 1 2
(x , y) x y -2 4 -1 1 2 3 4 5 6 7
-1.5 -2
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Quadratic Relations (Vertex Form): Transformations

7. Example 1: Parabola with Transformations
Find the vertex, the axis of symmetry, the direction of opening, x-intercept(s) and the y-intercept for the graph of the quadratic relation. State the mapping rule.
Multiply by C.D. 2
Vertex:
Opens up
Quadratic Relations (Vertex Form): Transformations
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8. Example 1: Parabola with Transformations
Find the vertex, the axis of symmetry, the direction of opening, x-intercept(s) and the y-intercept for the graph of the quadratic relation. State the mapping rule.
Vertex:
Vertical expands by factor of 2.
Opens down
Left by 3
Up by 4
Mapping Rule:
(x , y)  (x - 3, -2y + 4) x y
x - 3
-2y + 4 -2 4 -1 1 2
(x , y) x y -2 4 -1 1 2 -5 -4 -3 -2 -1 -4 2 4
Quadratic Relations (Vertex Form): Transformations
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9. Example 1: Parabola with Transformations
Find the vertex, the axis of symmetry, the direction of opening, x-intercept(s) and the y-intercept for the graph of the quadratic relation. State the mapping rule.
Opens down
Vertex:
Quadratic Relations (Vertex Form): Transformations
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10. Example 1: Parabola with Transformations
Find the vertex, the axis of symmetry, the direction of opening, x-intercept(s) and the y-intercept for the graph of the quadratic relation. State the mapping rule.
Vertex:
Vertical compresses by factor of 1/3.
Opens down
Left by 1
Down by 2
Mapping Rule:
(x , y)  (x - 1, -1/3 y - 2) x y
x - 1
-1/3y - 2 -2 4 -1 1 2
(x , y) x y -2 4 -1 1 2 -3 -2 -1 1
-3 1/3
-2 1/3 -2
Quadratic Relations (Vertex Form): Transformations
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11. Example 1: Parabola with Transformations
Find the vertex, the axis of symmetry, the direction of opening, x-intercept(s) and the y-intercept for the graph of the quadratic relation. State the mapping rule.
Multiply by C.D. 3
Vertex:
Opens down
Quadratic Relations (Vertex Form): Transformations
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12. Example 2: Parabola with Transformations
Determine a quadratic relation in vertex form which contains vertex (-4, -5) and passes through the point (2, -6). Determine the mapping rule for the transformations.
Mapping Rule:
Parabola compresses vertically by factor of ,
opens (concaves) down (Reflects about x-axis),
translates 4 units left and down 5 units.
Quadratic Relations (Vertex Form): Transformations
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13. Homework
Work sheet: Day 1: Quadratic Relations in Vertex Form (Transformations) Day 2: Graphing Quadratic Relations in Vertex Form Text: Check the website for updates
Quadratic Relations (Vertex Form): Transformations
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14. End of lesson Quadratic Relations (Vertex Form): Transformations
© 2017 E. Choi – MPM2D - All Rights Reserved