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Summary 4
Library of Functions
Important algebraic Functions in Mathematics.
1. Constant functions: y=c where c is a real number.



y
Properties:
The graph of a constant function is a horizontal line at y=c.
Domain: (-∞, ∞ )
Range: {c}
Examples: y=2, y=-3
Note: The inverse of y=c is x=c which is a vertical line, not a function.
y=c
x
2. Identity function: the linear function y=x is called the identity function.
Domain: ( ,  )
Range: ( ,  )
y
y=x
x
3. The absolute value function: y=|x|
Domain : ( ,  )
Range: [0.,  )
y
y=x
x
4. Linear functions: y=mx + b
 The graph is a straight line with slope m and y-intercept b.
 Domain: ( ,  )
Note: If m=0 we have a horizontal line (constant function)
Different forms of a linear relation
 Standard form: Ax + By = C or Ax + By + C =0
 Slope-intercept form (functional form) : y = mx + b or f(x) = mx + b
 Point-slope form: y  y1  m( x  x1 )
x y
  1 where a=x-intercept and b=y-intercept
 Intercepts form:
a b
 Horizontal line: y=k
 Vertical line: x=h
5. Functions
with absolute value: to graph a function of the form
y=|x-a| + |x-b| +c
Step 1. Find the values of y when x=a and x=b. They determine the braking points of the graph
Step 2. Graph the points found in part 1.
Step 3. Find the y value for any x value smaller than both a and b. Graph (x, y).
Step 4. Find the y value for any x value larger than both a and b. Graph (x, y).
Step 5. Sketch the graph through the given points.
-1-
6. Piece-wise functions: functions often defined with different equations for
differentparts of the domain.
 2 if x  3
x 1 if x  3
Example: y  
y
2
3
x
7. Direct Variation function: y = kx where k is a constant.
If y = kx, we say that y varies directly proportional to x.
8. Inverse variation function: y 
If y 
k
where k is a constant
x
k
, we say that y varies inversely as x.
x
9. The greatest integer function or bracket function: y=[x]. The greatest integer value of
a number x, denoted by [x] is given by the greatest integer that is less than or equal to the number.
Examples: [2.9]=2, [2.3]=2, [2]=2, [1.89]=1, [0.35]=0, [-2.78]=-3, [-0.05]=-1, [-3]=-3
y
Domain: ( ,  )
Range: {. . . ,-3,-2, -1, 0, 1, 2, 3, ...}
2
1
-2
-1
1
2
3
- -1
-2
10. The squaring function: the function y = x2 is called the squaring
function. The graph is the parabola shown below.
The graph of a quadratic equation is a parabola.
Example: y = x2 (the squaring function) is a parabola .
Domain: ( ,  ) and Range: [0,  )
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y
x
x
11. Quadratic Functions. Any function of the form y  ax 2  bx  x where
a, b, c are real numbers and a≠0 is a quadratic function
Every quadratic function y  ax 2  bx  x (general form) can be changed by using the
technique of completing the square to the standard form y  k  a( x  h) 2 ,where (h, k)
gives the vertex of the parabola and x = h is the line of symmetry. If a>0, the parabola
opens upward. If a < 0, the parabola opens downward.
2. The graph of y  ax 2  bx  x is the region below the parabola and the graph of
y  ax 2  bx  x is the region above the parabola.
Technique of completing the square to find the vertex of the parabola.
Function : y  ax 2  bx  c
Technique
Procedure
Step 1- Subtract c from both
members of the equation
Step 2- Factor “a” on the right
member
2
b2
1 b
Step 3  Add    
to
2 a
4a 2
b
x2  a
a
Example y  3x 2  6x  5
y  c  ax 2  bx
y  5  3x 2  6x
b 

y  c  a x 2  x 
a 

y  5  3 x 2  2x
yc


b2
b
b 2 
 a x 2  x 

4a
a
4a 2 


b2
36

1
4a 4(9)
y  5  3  3( x 2  2x  1)
2
b2
1 b
and a   
to y  c
4a
2 a
Step 4- use the fact that
2
2 
 2 b
 x  x  b    x  b 

a
2a 
4a 2  

yc
b2
b 

 a x  
4a
2a 

y  2  3( x  1) 2
2
Therefore, h = -1, k = 2
Vertex is (-1, 2)
Example 2. Find the vertex and the range of the parabola y  2x 2  7x  5
7 

Step 1 : y  2x 2  7 x  5  y  5  2x 2  7 x
Step 2 : y  5  2 x 2  x 
2 

2
49
1 7
Step 3 :    
16
 2 2
49
7
49 
49
7


Step 4 : y  5  2 
 2 x 2  x    y  5 
 2 x  
16
2
16 
8
4


2
9
7
7
9
9

7
 y   2 x    h  , k   , vertex is  ,  
8
4
4
8
8

4
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2