graphing quadratic functions 11.4 1.graph quadratic functions of the form f ( x ) = ax 2. 2.graph...
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Graphing Quadratic Functions11.411.4
1. Graph quadratic functions of the form f(x) = ax2. 2. Graph quadratic functions of the form f(x) = ax2 + k.3. Graph quadratic functions of the form f(x) = a(x – h)2.4. Graph quadratic functions of the form f(x) = a(x – h)2 + k.5. Graph quadratic functions of the form f(x) = ax2 + bx + c.6. Solve applications involving parabolas.
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Shape: Parabola
Axis: x = 0
Vertex: (0, 0)
x & y-intercepts:vertex
(0, 0)
axis of symmetry
x = 0 (y-axis)
2xxf
Imaginary Solutions
Perfect Square
Vertical Shifts
Vertex:
x = 0Axis:Vertex:
x = 0Axis:
Vertex:
x = 0Axis:Vertex:
x = 0Axis:
Graph: Graph:
Vertex:
32 xxf
Axis:
(0, -3)
x = 0
Axis is always x = x-coordinate of the vertex
Vertex:
x = 2Axis:
Vertex:
x = - 5Axis:
Vertex:
x = 3Axis:
Vertex:
x = - 4Axis:
Horizontal ShiftsOpposite of sign
Graph: Graph:
Vertex:
32 2 xxf
Axis:
(2, 3)
x = 2
opposite
same
Graph: Graph:
Vertex:
21 2 xxf
Axis:
(-1, 2)
x = -1
opposite
same
Slide 11- 8Copyright © 2011 Pearson Education, Inc.
What is the axis of symmetry for the function f(x) = (x + 3)2 + 5 ?
a) x = 3
b) x = –3
c) x = 5
d) y = –3
11.4
Slide 11- 9Copyright © 2011 Pearson Education, Inc.
What is the axis of symmetry for the equation f(x) = (x + 3)2 + 5 ?
a) x = 3
b) x = –3
c) x = 5
d) y = –3
11.4
If a > 1, the graph is narrower.
If 0 < a < 1, the graph is wider.
If a < 0, the graph opens downward.
If a > 0, the graph opens upward.
Graph: Graph: 41 2 xxf
Vertex: (-1, 4)
Direction: Down
Shape: Same
Axis: x = -1
axisrange
Vertex & 4 other points
y-intercept: (0, 3)
Mirrored point: (-2, 3)
x-intercepts:x = 1 (1, 0)
Mirrored point: (-3, 0) Range:
Domain: (- ∞, ∞)
(- ∞, 4]Pick a value for x:
Let x=0.
Too hard
Graph: Graph:
Vertex: (1, -3)
Direction: Up
Shape: Narrower
Axis: x = 1
axisrange
Vertex & 4 other points
y-intercept: (0, -1)
Mirrored point: (2, -1)
x-intercepts:
Let x = 3 (3, 5)
Mirrored point: (-1, 5)Range:
Domain: (- ∞, ∞)
[-3, ∞)
31x2xf 2
1x4xxf 2
Graph: Graph:
__x4x1xf 2
2x½ squared
2 5xf
52xxf 2
Vertex: (-2, -5)
44 + a = 1, b = 4, c = -1
a2b
x
124x 2
24
12422f 2
2fy
52f
Vertex: (-2, -5)
Vertex FormulaVertex Formula
Vertex of a Quadratic Function in the Form f(x) = ax2 + bx + c
1. The x-coordinate is .
2. Find the y-coordinate by evaluating .
a2b
x
a2b
f
ab
f,2ab
2Vertex:
f(x) = 3x2 – 12x + 4
122
6x Vertex: (2, 8)
2(2) 3(2) 12(2) 4f
(2) 12 24 4f
(2) 8f
Find the vertex: Find the vertex:
2ab
x
3
122
x
khxaxf 2 cbxaxxf 2
Read vertex from equation
Vertex: (opposite, same)
opposite
same
Use vertex formula
ab
f2a
b2
,
Slide 11- 18Copyright © 2011 Pearson Education, Inc.
What are the coordinates of the vertex of the function f(x) = x2 + 4x + 5?
a) (1, 2)
b) (0, 4)
c) (2, 1)
d) (4, 0)
11.4
Slide 11- 19Copyright © 2011 Pearson Education, Inc.
What are the coordinates of the vertex of the function f(x) = x2 + 4x + 5?
a) (1, 2)
b) (0, 4)
c) (2, 1)
d) (4, 0)
11.4
Slide 11- 20Copyright © 2011 Pearson Education, Inc.
What is the vertex of y = –2(x + 3)2 + 5?
a) (–3,5)
b) (3,–5)
c) (5,–3)
d) (2,–3)
11.4
Slide 11- 21Copyright © 2011 Pearson Education, Inc.
What is the vertex of y = –2(x + 3)2 + 5?
a) (–3,5)
b) (3,–5)
c) (5,–3)
d) (2,–3)
11.4
Graph: Graph: 862 xxxf
Vertex:(-3, -1)
Direction: Up
Shape: Same
Axis: x = -3
Vertex & 4 other points
y-intercept: (0, 8)
Mirrored point: (-6, 8)
x-intercepts:
Range:
Domain:
183633
316
2
fy
2x
(- ∞, ∞)
[ -1, ∞)
860 2 xx
(-2, 0) (-4, 0) 420 xx
42 xx
Let y=0.
Graph: Graph: 542 xxxf
Vertex:(-2, -1)
Direction: Down
Shape: Same
Axis: x = -2
Vertex & 4 other points
y-intercept: (0, -5)
Mirrored point:(-4, -5)
x-intercepts:
Range:
Domain:
152422
214
2
fy
2x
(- ∞, ∞)
(-∞, -1]Let x = -1 (-1, -2)
Mirrored point:(-3, -2)
None