Introduction to Quadratics
Objectives:
Define Quadratic Functions and Parent functions
Explore Parameter changes and their effects.
R. Portteus 2005-06
T. Merrill 2006
What is a Quadratic Function?
 A quadratic function is any function
whose graph is a parabola.
 A quadratic equation is any equation that
can be written in the form y = ax2 + bx + c.
The constants a, b, and c are called the
parameters of the equation. These
values tell us the shape and location of the
parabola.
Parent Functions
 The parent function is the simplest
function of a certain type. It is called this
because all of functions in that group look
like that and only change location and
shape.
 The linear parent function is y = x.
 All graphs are straight lines.
 The quadratic parent function is y = x2.
 All graphs are parabolas.
Effects on Parameters
 What happens to the graph of y = ax2
when a is changed?
 If a > 0, the parabola opens upward.
 If a < 0, the parabola opens downward.
Examples
y
y
x
y = 2x2 opens upward
x
y = -2x2 opens downward
Graph the following four parabolas
on the graph at right and label each
A.
B.
C.
D.
yx
y
2
y  x
2
y  2x
2
y  2x
2
x
Effect of –a
2
In a quadratic formula y  ax  bx  c , when a is
multiplied by -1 the resulting graph is the same as the original
Reflected over the x-axis
graph _____________________________.
Effects on Parameters
 If two quadratic functions of the form
y = ax2 have different coefficients, then
one graph will be wider than the other.
 Which of these functions produce the
widest graph?
 y = 3x2, y = -5x2, y = ¾ x2
Answer
y
y=
3x2
y = ¾ x2
y = ¾ x2 is wider, so
the smaller a is, the
wider the graph is.
x
y = -5x2
Graph the following four parabolas
on the graph at right and label each
A.
B.
C.
D.
y
1 2
y x
2
y  2x
2
y  4x
2
y  2x
2
Effect of |a|
2
In a quadratic formula y  ax  bx  c ,
narrower
when |a| increases, the resulting graph is ___________,
wider
when |a| decreases, the resulting graph is _____________.
x
Effects on Parameters
 If two quadratic functions of the form
y = (x - h)2 have different values for h then
one graph will be a translation right or left
from the other graph.
 Compare these three graphs:
y = (x – 0)2
y = (x + 1)2
y = (x – 4)2
Parent function!
•What happens to the parent function
when we “add 1”?
•What happens when we “subtract four”?
Answer
y
y = (x + 1)2 is translated left
one spaces from y = x2
y = (x – 4)2 is translated right
four spaces from y = x2
y = x2 - 4
y = (x+1)2
x
Effects on Parameters
 If two quadratic functions of the form
y = x2 + c have different constants, c, then
one graph will be a translation up or down
from the other graph.
 Compare the graphs of y = x2 + 3 with the
graph of y = x2 – 4.
Answer
y
y = x2 + 3
x
y = x2 - 4
y = x2 + 3 is translated up
three spaces from y = x2
and y = x2 – 4 is translated
down four spaces from y =
x2, so there are 7 spaces in
between the two graphs.
Graph the following four parabolas
on the graph at right and label each
A.
y
y  x x2
2
B.
y  x  x5
C.
y  x  x2
D.
y  x  x 5
2
x
2
2
Effect of c
2
In a quadratic formula y  ax  bx  c , when c increases
one unit, the resulting graph is __________________,
translated up 1 unit when c
down 1 unit
decreases one unit, the resulting graph is translated
__________________.
Examples
 Identify the parent function of the
following equations.
1. y = -3x2 + 4x – 7?
y = x2 → Quadratic Function
2. 3x – 4y = 8
y = x → Linear Function
Examples - Continued
Determine whether the function face upward
or downward.
1. y = -3x2 + 2
- Downward
2. y = ½ x2
- Upward
Examples- Continued
What is the new equation if the given
function is translated down 4 spaces?
1. y = x2 – 3

y = x2 – 7
2. y = x2 + 8

y = x2 + 4
Examples - Continued
 Order the equations from widest to most
narrow.
1. y = -6x2, y = ¼ x2, y = 4x2
y = ¼ x2, y = 4x2, y = -6x2
How do the two given equation compare?
2. y = -3x2 – 8 y = x2
Translated down 8 spaces, reflected over the x-axis
to face downwards and is narrower.
Lesson Check
 What causes a parabola to move left or right?
 What causes a parabola to move up or down?
 What causes a parabola to flip upside down?
 What causes a parabola to get wider?
 What causes a parabola to get narrower?