1. graph this piecewise function. f(x) = 3x – 1if x < -2 ½x + 1if x > -2 2. write an...
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1. Graph this piecewise function.
f(x) = 3x – 1 if x < -2 ½x + 1 if x > -2
2. Write an equation for thispiecewise function.
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Algebra II 1
Absolute Value
Algebra II
1.6 E
The graph of an absolute value function always forms a “V”!
Standard Form for Abs. Value
y = a|x – h| + kAlgebra II 3
y = a|x – h| + kVertex: (opposite of h, same as k)Line of Symmetry (LOS): x = opposite hReflections:
If “a” is negative the “V” will open down
If “a” is positive: the “V” will open upTransformations: (stretch & shrink)
If the |a| is bigger than one, then the graph will be skinnier.
If the |a| is less than one, the graph will be wider.
Algebra II 4
1. y = 3 |x – 5| + 1 2. y = ½ |x + 2| + 3
Vertex: (5, 1)
Vertex: (-2, 3)
Opens up Opens up
skinnier wider
Algebra II 5
3. y = -2 |x + 5| - 3 4. y = - |x – 3| + 3
Vertex: (-5, -3)
Vertex: (3, 3)
Opens down Opens down
Skinnier Same
Algebra II 6
1. Determine and graph the vertex
2. Graph the line of symmetry (LOS)
3. Make a table and graph the points
choose the next two x-values to the right of your vertex
4. Reflect your points over the LOS
5. Draw your “V”Algebra II 7
Graph:y = |x – 2| + 1
Opens up
Same width as |x|
Vertex: (2,1)
LOS: x = 2
Table
Reflect
x
3
4
y
2
3Algebra II 8
Graph:y = -½|x + 3| – 1
Opens down
Wider than |x|
Vertex: (-3,-1)
LOS: x = -3
Table
Reflect
x
-2
-1
y
-1.5
-2Algebra II 9
Graph:y = 3|x + 1| – 3
Opens up
Skinnier than |x|
Vertex: (-1,-3)
LOS: x = -1
Table
Reflect
x
0
1
y
0
3Algebra II 10
Graph:y = - |x – 3| + 3
Opens down
Same width as |x|
Vertex: (3,3)
LOS: x = 3
Table
Reflect
x
4
5
y
2
1Algebra II 11
1. Identify the vertex
2. Identify another point on the graph
3. Plug both the vertex and the point into y = a|x – h| + k (standard form)
4. Find “a” it is the only variable left!
5. Plug “a” and the vertex into standard form
y = a|x – h| + k (keep y and x here)
Algebra II 12
Write an absolute value function for: What is the vertex?
(1,0) this is h and k
(2, -2) is x and y
Plug these points in and solve for a.y = a|x – h| + k
-2 = a|2 – 1| + 0
-2 = a|1| + 0
-2 = a + 0
-2 = a
y = -2|x – 1| + 0Algebra II 13
Write an absolute value function for: What is the vertex?
(-2, -1) this is h and k
(-1, 2) is x and y
Plug these points in and solve for a.y = a|x – h| + k
2 = a |-1 + 2| – 1
2 = a |1| – 1
2 = a – 1
3 = a
y = 3|x + 2| – 1Algebra II 14
Write an absolute value function for: What is the vertex?
(-1, 2) this is h and k
(1,1) is x and y
Plug these points in and solve for a.y = a|x – h |+ k
1 = a|1 + 1| + 2
1 = a|2| + 2
1 = 2a + 2
-1 = 2a
-½ = a
y = -1/2 |x + 1| + 2Algebra II 15
Write an absolute value function for: What is the vertex?
(2, -2) this is h and k
(5, 0) is x and y
Plug these points in and solve for a.y = a|x – h| + k
0 = a|5 – 2| – 2
0 = a|3| – 2
2 = 3a
⅔ = a
y = ⅔|x – 2| – 2Algebra II 16
Algebra II
Real World Application
Algebra II
Real World Application
1. Graph: y = 2|x - 3| -1
2. Write an absolute value function of this graph:
y = -2| x – 1| + 4
Algebra II 19
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