Transcript Quadratic Functions

Exploring Quadratic
Graphs
Objective:
To graph quadratic
functions.
Parabola
The graph of any quadratic
function.
It is a kind of
curve.
Where are parabolas seen in the real
world?
The Arctic Poppy
Satellite Dishes
The Golden Gate Bridge
Trajectory
Headlights
Why is the parabola important?
 Suspension
Bridges use a
parabolic design
to evenly
distribute the
weight of the
entire bridge to
the supporting
columns.
Why is the parabola important?
 The Satellite Dish uses a parabolic shape to
ensure that no matter where on the dish
surface the satellite signal strikes, it is always
reflected to the receiver.
Why is the parabola important?
A car’s Headlights, and common flashlights, use
parabolic mirrors to project the light from the bulb
into a tight beam, directing the light straight out
from the car, or flashlight.
Standard Form
y=
2
ax
+ bx + c
Examples
y  5x
2
y  x 7 y  x  x 3
2
2
http://www.mathwarehouse.com/quadratic/parabola/interactiveparabola.php or http://www-groups.dcs.stand.ac.uk/~history/Java/Parabola.html
y = ax2 + bx + c
Positive “a” values mean the
parabola will open upwards and
will have a minimum.
point
Minimum point is also
called a vertex.
y = -ax2 + bx + c
Negative “a” values mean the
parabola will open downwards
and will have a maximum point
Maximum point is
also called a vertex.
Will the graph open up or down?
x  6x  4
2
Steps
1. Draw a table and insert
vertex of (0,0).
2. Choose two numbers greater than the
x coordinate and two numbers less.
3. Solve for Y Graph
y  2x
Ex.
2
X
2
1
2(1)  2
0
1
2
2(1)  2
2
2(2)  8
2x
2(2)  8
2
2
2
2
Y
8
2
0
2
8
(X,Y)
(2,8)
(1, 2)
(0, 0)
(1, 2)
(2,8)
(X,Y)
-2, 8
-1,2
0,0
1,2
2,8
Ex.
X
2
1
0
1
2
y  2 x +3
2
2
Y (X,Y)
2
2(2)  3  11 11 (2,11)
2
2(1)  3  5 5 ( 1,5)
3 (0,3)
2
2(1)  3  5
5 (1,5)
2
2(2)  3  11 11 (2,11)
2 x +3
(X,Y)
-2, 11
-1,5
0,3
1,5
2,11
Check It Out! Example 2b
Graph the quadratic function.
y = –3x2 + 1
x
y
–2
–11
–1
–2
0
1
1
–2
2
–11
Make a table of values.
Choose values of x and
use them to find values
of y.
Graph the points. Then
connect the points with a
smooth curve.
Ex.
f ( x)  2 x
2 x
X
2
2 2(2)  8
2
1 2(1)  2
2
0 2(0)  0
2
1 2(1)  2
2
2 2(2)  8
2
2
Y
8
2
0
2
8
(X,Y)
( 2, 8)
(1, 2)
(0, 0)
(1, 2)
(2, 8)
(X,Y)
-2,-8
-1,-2
0,0
1,-2
2,-8
Check It Out! Example 2a
Graph each quadratic function.
y = x2 + 2
x
y
–2
6
–1
3
0
2
1
3
2
6
Make a table of values.
Choose values of x and
use them to find values
of y.
Graph the points. Then
connect the points with a
smooth curve.
Additional Example 3A: Identifying the Direction
of a Parabola
Tell whether the graph of the quadratic
function opens upward or downward. Explain.
Write the function in the form
y = ax2 + bx + c by solving for y.
Add
to both sides.
Identify the value of a.
Since a > 0, the parabola
opens upward.
Additional Example 3B: Identifying the Direction
of a Parabola
Tell whether the graph of the quadratic
function opens upward or downward. Explain.
y = 5x – 3x2
y = –3x2 + 5x
Write the function in the
form y = ax2 + bx + c.
a = –3
Identify the value of a.
Since a < 0, the parabola opens downward.
Check It Out! Example 3a
Tell whether the graph of each quadratic
function opens upward or downward. Explain.
f(x) = –4x2 – x + 1
f(x) = –4x2 – x + 1
a = –4
Identify the value of a.
Since a < 0 the parabola opens downward.
Lesson Quiz: Part I
1. Without graphing, tell whether (3, 12) is on the
graph of y = 2x2 – 5. no
2. Graph y = 1.5x2.
Lesson Quiz: Part II
Use the graph for Problems 3-5.
3. Identify the vertex.
(5, –4)
4. Does the function have a
minimum or maximum? What is
it? maximum; –4
5. Find the domain and range.
D: all real numbers;
R: y ≤ –4