The graph of any quadratic function is a curve called a parabola.
Remember that any quadratic function is a transformation of the basic function y = x 2 .
Hopefully you remember from grade 10 that a quadratic can be written in transformational form.
1  y
 k
   x
 h

2 all variables are real numbers a a: vertical stretch
(h, k): coordinates of the vertex of the parabola
Vertical Stretch: a ratio that compares the change in y-values of a parabola with the corresponding y-values of y = x 2 .
Vertex of a parabola: the point on a parabola where a maximum or minimum value occurs.
Vertex at a maximum
 negative stretch.
Vertex at a minimum
 positive stretch.
Axis of Symmetry: A line in which a parabola or other graph is reflected onto itself. (For a parabola, this line will always run through the vertex)
Axis of Symmetry
Axis of symmetry has the equation x = h. For this graph, since the vertex is at (-3, 5) the equation for the axis of symmetry is x = -3.
Co-ordinates for vertex are (-3, 5)

The 3 Forms of the Quadratic Functions
1. Transformational Form: 1 y
 k
 x
 h
2 a
This form allows you to read the vertical stretch and find the vertex
Examples: a)
2
 y
  x

2

2
    easily. It also makes it easier to “see” the transformation of y = x 2 and help with the mapping notation of the function.
Vertical stretch: + ½
Vertex: (-2, 6)
Mapping: (x, y)  (x – 2, ½ y + 6)
b) 
1  y
 
2. Standard Form: x

5

2
3
Vertical Stretch: -3
Vertex: (5, -1)
Mapping: (x, y)  y
(x + 5, -3y – 1)
 a
 x
 h

2
 k
This form allows you to read the vertical stretch and find the vertex easily. This form also allows us to punch it into the TI-83 easily. (*Note: to change from transformational to standard form, simply use algebra to manipulate the equation for y.)
Examples:
  a)
2
 y

6
  x

2

2
 y

6
 
2 x

2

2
(Add 6) y

1
2
 x

2

2 
6
Vertical stretch : ½
Vertex: (-2, 6)

3. General Form y
 ax 2
 bx
 c
This form gives us the least amount of information about the properties of the parabola but it does give the vertical stretch and the y-intercept. It is the form that is given during regression analysis (on the TI-83) so it is still important. (*Note: to change from the standard form to the general form simply expand and simplify the equation)
Examples (from above): y

1
2
 x

2

2 
6
FOIL (x + 2) 2 y

1
2
 x
2 
4 x

4


6
Multiply bracket by ½ y = ½ x 2 + 2x + 2 + 6
Collect like terms y = ½ x 2 + 2x + 8
Stretch = ½ y-int @ 8
**We will learn (review) how to change from general form to transformational form soon by completing the square.**
Practice Question: -1/3(y + 1) = (x – 5) 2

Forms of Quadratic Functions – Practice
A.
Change each of the following equations into general form from its given transformational form.
1.
3(y – 2) = (x – 1) 2
2.
½ (y + 3) = (x – 5) 2
3.
-¼ (y + 1) = (x – 4) 2
4.
2/3(y + 2) = (x – 3) 2
5.
-2(y – 7) = (x + 10) 2
B.
For each of the following equations above state the:
vertical stretch
vertex
y-intercept
equation of axis of symmetry
C.
Sketch each of the graphs using the information in part B

Solutions:
Equation Equation (general) Stretch Vertex y-int Graph
(transformational)
1.
3(y – 2) = (x – 1) 2 y = 1/3x 2 – 2/3x +7/3 1/3 (1, 2) 7/3
2.
½ (y + 3) = (x – 5) 2 y = 2x 2 – 20x + 47 2 (5, -3) 47
3.
-¼ (y + 1) = (x – 4) 2 y = -4x 2 + 32x – 65 -4 (4, -1) -65

4.
2/3(y + 2) = (x – 3) 2 y = 3/2x 2 – 9x + 23/2 3/2 (3, -2) 23/2
5.
-2(y – 7) = (x + 10) 2 y = -½x 2 – 10x – 43 - ½ (-10, 7) -43