Download Properties of Graphs of Quadratic Functions

Properties of Graphs of Quadratic Functions
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 = x2.
Hopefully you remember from grade 10 that a quadratic can be
written in transformational form.
2
1
y  k    x  h

a
all variables are real numbers
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 = x2.
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
2
1
y  k    x  h

a
1. Transformational Form:
This form allows you to read the vertical stretch and find the vertex
easily. It also makes it easier to “see” the transformation of y = x2 and
help with the mapping notation of the function.
Examples:
a) 2 y  6   x  22
Vertical stretch: + ½
Vertex: (-2, 6)
Mapping: (x, y)  (x – 2, ½ y + 6)
b)  1  y  1   x  52
3
Vertical Stretch: -3
Vertex: (5, -1)
Mapping: (x, y)  (x + 5, -3y – 1)
y  a x  h   k
2. Standard Form:
2
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:
 2
(Add 6)
a) 2 y  6  x  2
2
 y  6  12 x  22
y  12 x  2  6
Vertical stretch : ½
Vertex: (-2, 6)
2
y  ax 2  bx  c
3. General Form
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  12 x  2  6
2
x

FOIL (x + 2)2
y
Multiply bracket by ½
y = ½ x2 + 2x + 2 + 6
Collect like terms
y = ½ x2 + 2x + 8
Stretch = ½
y-int @ 8
1
2
2
 4x  4  6
**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
(transformational)
1.
3(y – 2) = (x – 1)2
Equation (general)
y = 1/3x2 – 2/3x +7/3
Stretch
Vertex
y-int Graph
1/3
(1, 2)
7/3
2. ½ (y + 3) = (x – 5)2
y = 2x2 – 20x + 47
2
(5, -3)
47
3. -¼ (y + 1) = (x – 4)2
y = -4x2 + 32x – 65
-4
(4, -1)
-65
4. 2/3(y + 2) = (x – 3)2
y = 3/2x2 – 9x + 23/2
3/2
(3, -2)
23/2
5. -2(y – 7) = (x + 10)2
y = -½x2 – 10x – 43
-½
(-10, 7)
-43