Graphing Parabolas

1
Graphing Parabolas

• Using the Vertex
• Axis of Symmetry
• y-Intercept
• By Jeffrey Bivin
• Lake Zurich High School
• jeff.bivin_at_lz95.org

Last Updated October 15, 2007
2
Graphing Parabolas

• With your graphing calculator, graph each of the
  following quadratic equations and identify the
  vertex and axis of symmetry.

• y x2 4x - 7

y 2x2 10x 4
y -3x2 5x 9
Jeff Bivin -- LZHS
3
Graph the following parabola
x -2
y x2 4x - 7
vertex
axis of symmetry
(0, -7)
(-2, -11)
y-intercept
Jeff Bivin -- LZHS
4
Graph the following parabola
y 2x2 10x 4
vertex
axis of symmetry
y-intercept
Jeff Bivin -- LZHS
5
Graph the following parabola
y -3x2 5x 9
vertex
axis of symmetry
y-intercept
Jeff Bivin -- LZHS
6
Graphing Parabolas

• Now look at the coefficients of the equation and
  the value of the axis of symmetry especially a
  and b
• y ax2 bx c

• y x2 4x - 7

y 2x2 10x 4
y -3x2 5x 9
Jeff Bivin -- LZHS
7
Graphing Parabolas

• y ax2 bx c

Axis of symmetry
Vertex
Jeff Bivin -- LZHS
8
Graph the following parabola
x -2
re-visited
y x2 4x - 7
axis of symmetry
vertex
(0, -7)
(-2, -11)
y-intercept
Jeff Bivin -- LZHS
9
Graph the following parabola
re-visited
y 2x2 10x 4
(0, 4)
axis of symmetry
vertex
y-intercept
Jeff Bivin -- LZHS
10
Graph the following parabola
Why did this parabola open downward instead of
upward as did the previous two?
re-visited
y -3x2 5x 9
axis of symmetry
vertex
y-intercept
Jeff Bivin -- LZHS
11
Graph the following parabola
y x2 6x - 8
x -3
Axis of symmetry
(0, -8)
Vertex
(-3, -17)
y-intercept
Jeff Bivin -- LZHS
12
Graph the following parabola
y -2x2 7x 12
(0, 12)
Axis of symmetry
Vertex
y-intercept
Jeff Bivin -- LZHS
13
Graphing Parabolas

• In Vertex Form

Jeff Bivin -- LZHS
14
Graphing Parabolas

• With your graphing calculator, graph each of the
  following quadratic equations and identify the
  vertex and axis of symmetry.

vertex
axis of sym.

• y x2

y (x - 5)2 - 4
y -3(x 2)2 5
y ?(x - 3)2 1
Jeff Bivin -- LZHS
15
Graph the following parabola
x 5
y (x - 5)2 - 4
(0, 21)
axis of symmetry
vertex
(5, 4)
y-intercept
Jeff Bivin -- LZHS
16
Graph the following parabola
(-2, 5)
y -3(x 2)2 5
axis of symmetry
vertex
(0, -7)
x -2
y-intercept
Jeff Bivin -- LZHS
17
Graph the following parabola
x 3
y ?(x - 3)2 - 1
axis of symmetry
vertex
(3, -1)
y-intercept
Jeff Bivin -- LZHS
18
Graphing Parabolas

• In Intercept Form

Jeff Bivin -- LZHS
19
Graph the following parabola
x 1
0
y (x 4)(x 2)
x-intercepts
(4, 0)
(-2, 0)
(0, -8)
axis of symmetry
(1, -9)
vertex
y-intercept
Jeff Bivin -- LZHS
20
Graph the following parabola
x 5
0
y (x - 1)(x - 9)
(0, 9)
x-intercepts
(9, 0)
(1, 0)
axis of symmetry
(5, -16)
vertex
y-intercept
Jeff Bivin -- LZHS
21
Graph the following parabola
0
y -2(x 1)(x - 5)
(2, 18)
(0, 10)
x-intercepts
(5, 0)
(-1, 0)
axis of symmetry
x 2
vertex
y-intercept
Jeff Bivin -- LZHS
22
Convert to standard form
y -2(x 1)(x - 5)
y -2(x2 5x 1x 5)
y -2(x2 4x 5)
y -2x2 8x 10
Jeff Bivin -- LZHS
23
Now graph from standard form.
y -2x2 8x 10
(2, 18)
Axis of symmetry
(0, 10)
Vertex
x 2
y-intercept
Jeff Bivin -- LZHS
24
A taxi service operates between two airports
transporting 200 passengers a day. The charge is
15.00. The owner estimates that 10 passengers
will be lost for each 2 increase in the fare.
What charge would be most profitable for the
service? What is the maximum income?
VERTEX
Income Price ? Quantity
Define the variable
f(x) ( 15 2x ) ( 200 10x )
x number of 2 price increases
15 2x 0
200 10x 0
f(x) income
2x -15
200 10x
Vertex is
So, price (15 2x) (15 2(6.25)) 15
12.5 27.50
Maximumincome
27.50
137.50
Jeff Bivin -- LZHS
25
A taxi service operates between two airports
transporting 200 passengers a day. The charge is
15.00. The owner estimates that 10 passengers
will be lost for each 2 increase in the fare.
What charge would be most profitable for the
service? What is the maximum income?
Alternative Method
VERTEX
Income Price ? Quantity
Define the variable
f(x) ( 15 2x ) ( 200 10x )
x number of 2 price increases
f(x) 3000 150x 400x 20x2
f(x) income
f(x) 20x2 250x 3000
f(6.25) 20(6.25)2 250(6.25) 3000
f(6.25) 3781.25
Vertex is
So, price (15 2x) (15 2(6.25)) 15
12.5 27.50
Maximum income f(x) 3781.25
Jeff Bivin -- LZHS