Graphing Quadratic Functions
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Graphing Quadratic Functions
Graphs of Quadratic Functions
Vertex
Axis of symmetry
12
10
8
6
4
2
-2
-10 -5 5 10
g x = -0.5x+4 2+6
f x = 2x-2 2+1
x-intercepts
Important features of graphs of parabolas
Graphing QuadraticsIf you were asked to graph a quadratic, what information would you need to know to complete the problem?
The vertex, because we need to know where the graph is located in the plane
If the parabola points up or down, and whether it opens normal, narrow or wide
Our graphs will be more “quick sketches” than exact graphs.
10
8
6
4
2
-2
-4
-10 -5 5 10
f x = x2
Graph of f(x)=x2
x f(x)
0
1
-1
2
-2
0
1
1
4
4
Axis is x = 0
Vertex at (0, 0)
Points up, opens “normal”
Notice the symmetry
More with Vertex Form
The vertex is (h, k). Changes in (h, k) will shift the quadratic around in the plane (left/right, up/down).
The axis of symmetry is x = h
Notice that you take the opposite of h from how it is written in the
equation
Example #1
Example #2
2( ) ( )f x a x h k
If a > 0, the graph points upIf a < 0, the graph points down
2( ) ( 4)f x x
2( ) 6f x x
Example #32( ) 5( 3) 1f x x
Vertex is _____
Axis is _______
Points _______Vertex is _____Axis is _______Points _______
Vertex is _____Axis is _______Points _______
Vertex is (0, 6)
Axis is x = 0
Points up
Vertex is (-3, -1)
Axis is x = -3
Points up
Vertex is (4, 0)
Axis is x = 4
Points down
Equations of Quadratic Functions
Vertex Form Standard Form
2( ) ( )f x a x h k 2( )f x ax bx c
More with Standard Form
To find the x-value of the vertex, use the formula
To find the y-value, plug in x and solve for y
The axis of symmetry is
Example #1
2( )f x ax bx c
If a > 0, the graph points upIf a < 0, the graph points down
2( ) 4 4f x x x
Vertex is _____
Axis is _______
Points _______
b = 4, a = -1
2
bx
a
2
bx
a
42
2( 1)x
Find x-value of vertex using formula
Find y-value using substitution2( ) 4 3f x x x 2 (2) (2) (2) 1f
Vertex is (2, 1)
Axis is x = 2
Points down
More examplesExample #2
2( ) 3 5f x x x
Vertex is _____
Axis is _______
Points _______
b = -1, a = 3
( 1) 1
2 2(3) 6
bx
a
Find x-value using formula
Find y-value using substitution
2( ) 3( ) 5f x x x 16
1 59
6 12f
Vertex is (1/6, 59/12)
Axis is x = 1/6
Points up
16
16
You try:
Vertex is _____Axis is _______Points _______
2( ) 2 12 1f x x x
Vertex is (3, 17)
Axis is x = 3
Points down
More about a10
8
6
4
2
-5 5
g x = x2
10
8
6
4
2
-5 5
g x = 5x2a =1
a =1/5
a = 5
If a is close to 0, the graph opens _______________If a is farther from 0, the graph opens ____________If a > 0, the graph points________ If a < 0, the graph points ________
10
8
6
4
2
-5 5
g x = 0.2x2
What happens to the graph as the value of a changes?
If a is close to 0, the graph opens widerIf a is farther from 0, the graph opens narrowerIf a > 0, the graph points up If a < 0, the graph points down
When a = 1, the graph is “normal”
Graphing QuadraticsIf you were asked to graph a quadratic,
what information would you need to know to complete the problem?
The vertex, because we need to know where the graph is located in the planeThe value of a, because we need to know if it points up or down, and whether it opens normal, narrow or wide
Our graphs will be more “quick sketches” than exact graphs.
Sketch each quadratic2( ) 5( 3) 1f x x
2( ) 4 3f x x x
2( ) 8 18f x x x
6
4
2
-2
-4
-6
-5 5
6
4
2
-2
-4
-6
-5 5
6
4
2
-2
-4
-6
-5 5
6
4
2
-2
-4
-6
-5 5
21( ) 4
2f x x
V = (0, 4)Points downWide
V = (-4, 2)Points upNormal
V = (2, 1)Points downNormal
V = (-3, -1)Points upNarrow
Finding x-intercepts of quadratic functions
What are other words for x-intercepts?
Name 4 methods of finding the x-intercepts of quadratic equations:
All are the value of x when y = 0
Summary: Be able to compare and contrast vertex and standard form
Vertex Form Standard Form
How do you find the Vertex?
How do you find the Axis of Symmetry?
How can you tell if the function:• points up or down? •opens normal, wide or narrow?
What info is needed to do a quick sketch or graph?
How do you find the solutions? (x-intercepts, roots, zeroes, value of x when y = 0)
Set = 0, get “squared stuff” alone, then use square root method
Set = 0 and use method of choice (factor, formula or square root)
2( ) ( )f x a x h k 2( )f x ax bx c
Max and Min ProblemsWhat is the definition of the
maximum or minimum point of a quadratic function?
If a quadratic points down, the vertex is a maximum point
The vertex of a quadratic function is either a maximum point or a minimum point
max
min
If a quadratic points up, the vertex is a minimum point
If you are asked to find a maximum or minimum value of a quadratic function, all you need to do is find its vertex
Example
An object is thrown upward from the top of a 100 foot cliff. Its height in feet about the ground after t seconds given by the function f(t) = -16t2 + 8t + 100.
What was the maximum height of the object?
How many seconds did it take for the object to reach its max height?
How can we find the answer? What is the question asking for?
Example
What was the maximum height of the object?
How many seconds did it take for the object to reach its maximum height?
What is the definition of the maximum or minimum point of a quadratic function?
The vertex of a quadratic function is either a maximum point or a minimum point vertex
ExampleStep 1: Visualize the problem
f(t) = -16t2 + 8t + 100.To find the max values, find the vertex
The x-value of the vertex is the max time
The y-value of the vertex is the max height
Output: heightInput: time
8 8 1
2( 16) 32 4
It took about .25 seconds for the object to reach its max height
f(t) = -16t2 + 8t + 100.f(1/4) = -16(1/4)2 + 8(1/4) + 100.
f(1/4) = 101 The max height was 101 feet
(1/4, 101)
Step 2: Understand the equation
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