Notes: Vertex Form, Families of Graphs, Transformations
I. Families of Graphs
Families of graphs: a group of graphs that displays one or more characteristics
Parent graph: A basic graph that is transformed to create other members in a family of graphs.
The parent graph of a quadratic function is y = x2.
All quadratic functions are in the family of y = x2.
If we know what the parent graph looks like, we can use
transformations to graph any graph in that family.
Transformations include reflections, translations
(both vertical and horizontal) , expansions, contractions,
and rotations.
Key Points: (-3, 9), (-2, 4),
(-1, 1), (0,0), (1, 1), (2, 4),
(3, 9).
II. Vertex Form and Transformations
A. Vertex form is the form of the quadratic equation that will allow us to use transformations to graph.
TIPS
For vertical translations,
add ‘k’ to all y values.
For horizontal translations,
add ‘h’ to all x-values.
For reflections over x-axis,
take the opposite of all yvalues.
For stretches, multiply all y
values by |𝑎|.
B. Translations:
Whenever we translate a function, this means we pick up each point of the parent graph and shift it in the desired
direction for the desired number of units.
Algebraically this means that we add h to each x-value. For example, if h is -2, (3, 9) becomes (1, 9).
Algebraically, this means we add k to each y-value. For example, if k = 2, (3, 9) becomes (3, 11).
C. Expansions, Contractions, Reflections
Reflections:
You have done reflections before- when graphing quadratic equations, you reflect points across the axis of symmetry to
find more points. A reflection can be over any line, most often the x-axis or the y-axis.
In vertex form, if a is negative, all points are reflected over the x-axis.
On the graph to the left, the purple graph is
y = x2. The red graph is y = -x2 has been
reflected over the x-axis. Algebraically, you
take the opposite of all y-values, e.g. (3, 9)
becomes (3, -9).
Expansions and Contractions
Expansions and contractions change not only the location of the graph, but its size as well. Think of an object in Word or
PowerPoint. You can resize it by dragging a corner in and out. The shape stays the same, but the size is smaller or larger.
For a quadratic function in vertex form, if 0 < |𝑎| < 1, the graph is stretched by a factor of a. This means you multiply all
1
y-values by a. For example, For example, if a = 2, (2, 4) becomes (2, 2). This makes the graph look wider. If a = 2, (2, 4)
becomes (2, 8). This makes the graph look thinner.
V. Putting it all Together: Graphing Using Vertex Form and Transformations
When you have multiple transformations in one equation, you must perform them in a particular order if you want to
end up with the correct graph. The correct order is:
1) Horizontal Shifts
2) Stretch/Compress
3) Reflections
4) Vertical Shifts
Examples: See hand-written notes.