Ch1 - Algebraic and Non-Algebraic Functions and Their Inverse

CHAPTER

1

−

Algebraic and Non Algebraic Functions and Their Inverse 

 





Equations and their graphs  Lines, Circles, Parabolas; Ellipses, Hyperbolas, and Conic sections. Functions; Domain, Range, and their graphs. Algebraic functions  Piecewise-defined functions, Linear functions, Power functions, Polynomial functions, Increasing and decreasing function, Rational functions, and Even and Odd functions. Non Algebraic or Transcendental functions  Trigonometric functions, Inverse Trigonometric functions, Logarithmic functions, Exponential functions, Hyperbolic functions, and Inverse H yperbolic functions. Shifting a Graph of a Function.

−

Equations and Their Graphs

 

The graph of an equation involving and all points { , ) satisfying the equation.





LINES

−  

as its only variables consists of

−

The equation = + is called the slope intercept equation of the line with slope m and y and y intercept b.

−   −  −

− −  −

Example (1): Find the slope and y and y intercept of the line: 2

= 3?

The equation is equivalent to =2 3, which is the slope intercept equation of the line with slope = 2 and у intercept = 3.

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Mathematics I / 1 . Semester / Dr. / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions



CIRCLES

A circle is the set of points in a plane whose distances (radius) from a fixed point in the plane plane is constant. 

The general equation of a circle with center at (h ( h, k ) is:

 −   −      +



=





( , )

The equation of a circle with its center at the origin is

+



ℎ

( , )

=







( , )





(0,0)





ℎ

( , 0)

 

(0, )

Example (2):

 −         − ℎ  −   −     − −   −     −  −   −  −      −  − 1

(a) Find the center and radius of the circle: 2

Comparing with: h =1 , k = 5 (b) If the circle:

2

and

+

2

+ a = 3

=

2

2

+

+5

2



=3

shows that:

2

= 25 is shifted 2 units to the left and 3 units up, find its

3

2

new equation?

( 2 )2 +

+2

2

+

3

2

= 25

= 4 ,

So c is ( 2,3)

+ 2 2 +4 4=0? Completing the square shows that the given equation is equivalent to the 1 2+ + 2 2 = 9. Hence, its graph is the circle with center equation (1, 2) and radius 3.

(c) What is the graph of the equation:

2

2

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

CIRCLES

A circle is the set of points in a plane whose distances (radius) from a fixed point in the plane plane is constant. 

The general equation of a circle with center at (h ( h, k ) is:

 −   −      +



=





( , )

The equation of a circle with its center at the origin is

+



ℎ

( , )

=







( , )





(0,0)





ℎ

( , 0)

 

(0, )

Example (2):

 −         − ℎ  −   −     − −   −     −  −   −  −      −  − 1

(a) Find the center and radius of the circle: 2

Comparing with: h =1 , k = 5 (b) If the circle:

2

and

+

2

+ a = 3

=

2

2

+

+5

2



=3

shows that:

2

= 25 is shifted 2 units to the left and 3 units up, find its

3

2

new equation?

( 2 )2 +

+2

2

+

3

2

= 25

= 4 ,

So c is ( 2,3)

+ 2 2 +4 4=0? Completing the square shows that the given equation is equivalent to the 1 2+ + 2 2 = 9. Hence, its graph is the circle with center equation (1, 2) and radius 3.

(c) What is the graph of the equation:

2

2

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

PARABOLAS

      ≠   −  ……………………………  −  −

The general form of parabola equation is = + + . The graph of this 0 is a parabola. The parabola opens upward if > 0 and equation when downward if < 0. The axis is the line



=

−

..(1) 2 The vertex of the parabola is the point where the axis and parabola intersect. Its

x coordinate is

=

2

; and its y its y coordinate is found by substituting

in the parabola’s equation.

 −  =

2

Besides determining the direction in which the 2 parabola = opens, the number a is a scaling factor. The parabola widens as a approaches zero and narrows as becomes large.

   

Sketch the graph of

   =

, see Fig.(1.1):



1. Make a table of pairs that satisfy the function (substitute few values of and calculate the associated values of ). 2. Plot the points of ( x, x, y) y) appear in the table. 3. Draw a smooth curve through the plotted points. These points suggest a curve, which belongs to a family of curves called parabolas.



3

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Mathematics I / 1 . Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions

Fig.(1.1): The graph of

  =

2

.

In Fig.(1.1), we note that: The graph of = 2 contains the origin (0, 0) with all its points lie above the axis, When are positive and increasing, increases without bound. Hence, in the first quadrant, the graph moves up without bound as it moves right. 2 Since = 2 , it follows that, if any point , lies on the graph in the first quadrant, then the point , also lies on the graph in the second quadrant. Thus, the graph is symmetric with respect to the axis. The axis is called the axis of symmetry of this parabola. 





 

 − 

 − 

 

 −    −      =

Sketch the graph of





 −  −

If , is on the graph of the parabola = 2 (shown in Fig.1.1), then , 2 2 , and vice versa. Hence, the graph of = is on the graph of = is the 2 reflection in the axis of the graph = . The result is the parabola shown in Fig.(1.2).

−− − −− − Fig.(1.2): The graph of 4

 − =

2

.

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   −       −  =

Sketch the graphs of

=

and

1. The graph of = 2 is obtained from the parabola = 2 (shown in Fig.1.1) by exchanging and . The resulting curve is a parabola with the axis as its axis of symmetry and its "nose" at the origin (see Fig.(1.3b)). 2 2. The graph of = is the reflection in the result is the parabola shown in Fig.(1.3a).

 − =

axis of the graph



2

  =

   =

2

. The

2

 (b)

(a) Fig.(1.3): The graphs of

   − 2

=

and

 −  −        −  − − −  −  −  − − −−  ∴ − − −  −  − −     − Example (3): Graph the equation

=

Comparing the equation with

= 4. Since =

1

=

2

+2 2

= 2,

2

+

=

2

=

( 1)

2( 1/2)

=

1

1, we have

( 1)2

1 +4=

9

2

The vertex is ( 1, 9/2) The x intercepts (put = 0): 1 2 +4=0 2 2

2

+

2

.

+ 4?

shows that

 −  − =

1 2

,

=

1,

< 0, the parabola opens downward. From Equation (1) the axis is

the vertical line When

1

=

=

8=0

+4 =0

=

4

Plot some points, and sketch the axis, complete the graph shown in Figure.

5

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

ELLIPSES

 + 2

To sketch the graph of the equation:

9

2

= 1, compute a few values and

4

plot the corresponding points, as shown in Fig.(1.4). The graph suggested by these

  points belongs to a family of curves of the form ( +  

 + 2


 + 2

Now to graph 





− ≤ −  

Find

9

2

9

2

2

2

2

2

4

= 1) called ellipses.

= 1.

= 1:

 −   ≤   −   −  −  −     − 4

intercept (by putting

= 0), and since

2

9

2

9

+

2

4

= 1, it follows

that 2 9 , and therefore, = 3 and = 3. Its rightmost point is (3, 0), and its leftmost point is ( 3, 0). Find intercept (by putting = 0) gives = 2 and = 2, and that its lowest point is (0 , 2) and its highest point is (0, 2). In the first quadrant, as increases from 0 to 3, decreases from 2 to 0. If ( , ) is any point on the graph, then ( , ) also is on the graph. Hence, the graph is symmetric with respect to the axis. Similarly, if ( , ) is on the graph, so is ( , ), and therefore the graph is symmetric with respect to the axis.



 +  = 1 is a circle When a = b, the ellipse   with the equation  +  =  , that is, a circle 2

2

2

2

2

2

2

with center at the origin and radius a. Thus, circles are special cases of ellipses. The standard equation of an ellipse with center at (h, k ) is

−ℎ  −  2

2

2

+

2

Graph of the ellipse

=1

 +   2

2

2

2

= 1, a > b,

where the major axis is horizontal. 6

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

HYPERBOLAS

 − Consider the graph of the equation: 2

9

2

4

= 1. Some of the points on this

graph are tabulated and plotted in Fig.(1.5). These points suggest the curve shown

  in the figure, which belongs to a family of curves of the form ( −   hyperbolas.

 − 2


 −  = 1: Now to graph  = 1 +  ≥ 1, it follows that  Since 2

2







9

9

9

2

2

2

2

= 1) called

2

4

= 1.

2

2

4

4

   ≥  −  −      −     −   2

≥

9, and therefore,

3. Hence,

there are no points on the graph between the vertical lines = 3 and = 3. If ( , ) is on the graph, so is ( , ); thus, the graph is symmetric with respect to the axis. Similarly, the graph is symmetric with respect to the axis.

 

Note in Fig.(1.5) ; the dashed lines ( =

2 3

and,

=

2 3

) are called the

asymptotes of the hyperbola: Points on the hyperbola get closer and closer to these asymptotes as they recede from the origin. In general, the asymptotes of the hyperbola



 −   2

2

2

2

= 1 are the lines

=

and

=

.

CONIC SECTIONS

Parabolas, ellipses, and hyperbolas together make up a class of curves called conic sections. They can be defined geometrically as the intersections of planes with the surface of a right circular cone, as shown in Fig.(1.6). 7

Fig.(1.6): Conic sections.

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Functions and Their Graphs 

Functions; Domain and Range

A function is defined as a set of ordered pairs ( x, y), such that for each value of the independent variable x, there corresponds only one value of the dependent variable y. The set of x values is called the domain of the function, while the set of all corresponding values of is called the range of the function. The notation f ( x) is often used in place of y to indicate the value of the function f for a specific x and is read “ f of x” or “ f at x.”



Example (4): Verify the domains and ranges of these functions.

Solution: 

  −∞ ∞

−   ∞      ≥   −        −  ≥     ∞

The formula = 2 gives a real y value for any real number x, so the , ). The range of = 2 is [0, ] because the square of any domain is( real number is nonnegative and every nonnegative number y is the square of its own square root,





=

2

for

0.

The formula = 1/ gives a real y value for every x except = 0. we = 1/ , the set of cannot divide any number by zero. The range of reciprocals of all nonzero real numbers, is the set of all nonzero real numbers, since = 1/(1/ ).

0. The range of The formula = gives a real y value only if = is [0, ] because every nonnegative number is some number’s square root (namely, it is the square root of its own square). 8

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



 − −  ≤ ∞ − −  −

≤

0, or 4

− − −

2 gives a real y value for every x in the closed The formula = 1 2 interval from 1 to 1. Outside this domain, 1 is negative and its square 2 root is not a real number. The values of 1 vary from 0 to 1 on the given domain, and the square roots of these values do the same. The range of 2 is [0, 1]. 1

 − 

−≥ −

= 4 , the quantity 4 cannot be negative. That is, 4 4. The formula gives real values for all 4. The range of is [0, ], the set of all nonnegative numbers. In

Graphs of Functions

          

The graph of a function is the graph of the equation: = ( ). If ( x, y) is a point on the graph, then = ( ) is the height of the graph above the point x if ( ) is positive or below x if ( ) is negative (see Fig.(1.7)). Fig.(1.7): If ( x, y) lies on the graph of f , then the value = ( ) is the height of the graph above the point x (or below x if ƒ( x) is negative)..

  

Example (5): Graph the function

  =

+ 2 and find its domain and range.

Solution:

  

The graph of is the graph of the equation = + 2, which is the straight line with slope 1 and intercept 2. The set of all real numbers is both the domain and range of , (see Fig.(1.8)).





 

= + 2 is the set Fig.(1.8): The graph of of points ( x, y) for which y has the value x + 2. 9

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−

Example (6): Graph the function over the interval [ 2, 2] Solution:

−



1. Make a table of pairs that satisfy the function, in this case: y = 2 . 2. Plot the points ( x, y) whose coordinates appear in the table. 3. Draw a smooth curve through the plotted points, and label the curve with

its equation.

Example (7): The following are some examples of equations that are functions. (a) (b) (c) (d) (e)

  

 − −−

=3 +1 = 2 = 5 = 3 =



(f) (g) (h) (i)

3

2 +4

      

= 2 +9 =

6

= tan = cos 2

Example (8): The following are some examples of equations that are not functions; each has an example to illustrate why it is not a function.

(a) (b) (c) (d) (e) (f)

    −  − 

= 2 = +3 = 5 2 + 2 = 25 =± +4 2 2 =9

If If If If If If

  −  −

= 4, = 2, = 5, = 0, = 5, = 5,

then then then then then then 10

 −     

−− −− −

=2 = 2. or = 5 or = 1. can be any real number. =5 or = 5. = +3 or = 3. =4 = 4. or

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Algebraic Functions An algebraic function is a function constructed from polynomials using algebraic operations (addition, subtraction, multiplication, division, and taking roots). Fig.(1.9) displays the graphs of three algebraic functions.

(a)

(b)

(c)

Fig.(1.9): Graphs of three algebraic functions.



Piecewise-Defined Functions

These functions are described by using different parts of its domain, such as the absolute value function.

      ≥ −  −  ≤ Example (9): Graph the function

and find its domain and range.

    ∞

= = = when The graph of is shown in Fig.(1.9). Notice that 0, whereas, = 0. the domain of consists of all real when numbers ( ∞, ∞) , but the range is the set of all nonnegative real numbers [0, ].

Fig.(1.9): The absolute value function has domain ( 11

−∞ ∞ ,

) and range [0,

∞

].

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Example (10): Graph the following function over the interval [0, 1]

−     ≤ ≤   −     ,

=

2

<0

,

0

1,

Solution:

1

>1

≤ ≤

The values of ƒ are given by; = when < 0, = 2 when 0 1, and = 1 when > 1. The function, however, is just one function whose domain is the entire set of real numbers (see Fig.1.10).



  

Fig.(1.10): To graph the function = ( ) shown here, we apply different formulas to different parts of its domain.

Homework (1): 

Find the domain and range of each of the following functions:

12

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

−

Graph the following piecewise defined functions: 1.

2. 3. 4. 5.



Linear Functions

      

A function of the form = + , for constants m and b, is called a linear function. Fig.(1.11) shows an array of lines = where = 0, so these lines pass through the origin. Constant functions result when the slope = 0 (see Fig.(1.12)).

 

Fig.(1.11): The collection of = lines has slope m and all lines pass through the origin. 13

Fig.(1.12): A constant function has slope = 0.



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Power Functions



       − 

A function = where a is a constant, is called a power function. There are several important cases to consider.

(a)

= , a positive integer.

The graphs of = , for =1, 2, 3, 4, 5, are displayed in Fig.(1.13). These functions are defined for all real values of x. Notice that as the power n gets larger, the curves tend to flatten toward the x axis on the interval ( 1, 1), and also >1. Each curve passes through the point (1, 1) and rise more steeply for through the origin.

−

   −∞  ∞  −   − −     ≠ −      =

Fig.(1.13): Graphs of

(b)

=

,

= 1, 2, 3, 4, 5 defined for

<

<

.

=

= 1 = 1/ and g = 2 = 1/ 2 are The graphs of the functions shown in Fig.(1.14). Both functions are defined for all 0. The graph of = 1/ is the hyperbola = 1 which approaches the coordinate axes far from the origin, and the graph of y 1/ 2 also approaches the coordinate axes. =

Fig.(1.14 ): Graphs of



for part (a) and for part (b). 14

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             ∞         (c)

=

,

,

, and

3

= 1/2 = = 1/3 = The functions and g are the square root and cube root functions, respectively. The domain of the square root function is [0, ], but the cube root function is defined for all real x. Their graphs are 3/2 2/3 displayed in Fig.(1.15) along with the graphs of y and y . (Recall that =

3/2

=

1/2 3

and

2/3

=

1/3 2

Fig.(1.15): Graphs of

=

.)

    =

,

15

=

 ,  , ,   

and

. 

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

Polynomials

A function p is a polynomial if:

      ⋯…… − −       −∞ ∞  ≠      ≠         =

0

+

1

+

2

2

+

+

1

1

+

where, n is a positive integer and the numbers 0, 1, 2, …….., are real constants (called the coefficients of the polynomial). All polynomials have domain ( , ). If the leading coefficient 0 and > 0, then n is called the degree of the polynomial. Linear functions; = + , with 0 are polynomials of degree 1. 2 = + Quadratic functions are polynomials of degree 2 and written as, + . Likewise, cubic functions are polynomials of degree 3 and written as, 3 = + 2+ + . Fig.(1.16) shows the graphs of three polynomials.

Fig.(1.16): Graphs of three polynomial functions.



Increasing and Decreasing Functions

If the graph of a function rises as you move from left to right, we say that the function is increasing , and if the graph falls as you move from left to right, the function is decreasing . Some examples of these functions are shown in Fig.(1.17).

16

st


Fig.(1.17): Graphs of increasing and / or decreasing functions.

17

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

Rational Functions

  

A rational function is a quotient or ratio of two polynomials: ( ) = ( ) where, p and q are polynomials. The domain of a rational function is the set of all real x for which, ( ) 0.

 ≠

For example, the function

  −  ≠ −  =

2

2

3

7 +4

4/7 . Its graph is shown in is a rational function with domain Fig.(1.18a) with graphs of two other rational functions in Figs.(1.18b and 1.18c).

Fig.(1.18): Graphs of three rational functions. 

Even Functions and Odd Functions

18

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

   =

2

( , )

  −

( ,

)

   

 

(a): Graphs of = 2 and = 2 (an even functions) are symmetric about the y and x axis, respectively.

−

(b): Graph of = 3 (an odd function) is symmetric about the origin.

Fig.(1.19 ): Graphs of even and odd functions.

Example (11): Test whether the following functions are even, odd, or neither ? 







    = =

    = =

−   −  

Even function: +1 Even function: (see Fig.1.20a).

2

=

2

2

+1 =

−

for all ; symmetry about y axis. 2

− −  − −≠ − ≠ −− −

Odd function: +1 Not odd: Not even:

=

= +1

−

+ 1 for all ; symmetry about y axis.

for all ; symmetry about the origin.

+ 1 , but + 1 for all

= 1. The two are not equal. 0 (see Fig.1.20b).

Fig.(1.20): Even, odd and neither functions for Example (9). 19

st


Non-Algebraic or Transcendental Functions 

Trigonometric Functions

The six basic trigonometric functions are:

                 − ,

,

=

=

,

1

=

,

=

1

,

=

1

These functions are defined using a circle with equation 2 + 2 = 2 and the angle in standard position as shown in Fig.(1.21) with its vertex at the center of the circle and its initial side along the positive portion of the x axis.

Fig.(1.21): Defining of trigonometric

 

Functions in terms of , and .

−

Fig.(1.22): Angles in standard position in the xy plane. 20

st


   

o

The variable is generally expressed in radians (π radians = 180 ). For real values of , and lie between 1 and 1 inclusive.

−

The following are some properties of these functions:

            − − −−  −                ∓      ∓   −    −   −     2

2

+

2

1+ 1+

=1 2

= =

2

2

= = =

+2 = +2 = + =

( ± )= ±

=

±

=

2 = 2 2 =

2

2

2

2

=

1

=

±

±

1

2

2

=2

2

1

=1

−   2

cos

2

B

2

2

The relationship between the angles and sides of a triangle may be expressed using the Law of Sines or the Law of Cosines (see Fig.1.23). 21





1+cos

A



Fig.(1.23): Relations between sides

and angles of a triangle.

C

st


Laws of Sines:

Laws of Cosines:

 = =    

   −      −      −   2

=

2

+

2

2

cos

2

=

2

+

2

2

cos

2

=

2

+

2

2

cos

The graphs of the six trigonometric functions are shown in Fig.(1.24)

Fig.(1.24): Graphs of the (a) cosine, (b) sine, (c) tangent, (d) secant, (e) cosecant, and (f) cotangent functions using radian measure. The shading for each trigonometric function indicates its periodicity. 22

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

Inverse Trigonometric Functions

     

(a) (b) (c) (d) (e) (f)



= = = = = =

−−   −  −   −    − 

− ≤≤ ≤≤ −− ≤≤  −  − ≤≤   ≤≤ −−  ≤≤

1

,

1

, (0

1

, (

/2

1

=

1

1

=

1

=

(

/2

)

/2)

1/ ,

1

2

/2)

/2

(

1/ ,

(0

)

1

, (0

)

Logarithmic Functions



/2)



≠

These functions are of the form, = log , where the base 0, 1 is a positive constant. These and the exponential functions are inverse functions. If a = e = 2.71828….called the natural base of logarithms, we write = log = ln , called the natural logarithm of x. Fig.(1.25) shows the graphs of four logarithmic functions with various bases. In each case the domain is (0, ) and the , . range is



 −∞ ∞

∞



Fig.(1.25): Graphs of four logarithmic functions.



Exponential Functions

 

≠ −∞ ∞

These are functions of the form, = , where the base 0,1 is a positive constant. All exponential functions have domain ( , ) and range 0, . The graphs of some exponential functions are shown in Fig.(1.26).

 ∞

23

st


Fig.(1.26): Graphs of exponential functions.



Hyperbolic Functions

ℎ ℎ ℎ

(a) (b) (c)

= = =

  − −    − ℎ  =   −−− ℎ   

(d)

2

+

(e)

2

(f)

+

ℎ ℎ ℎ

= = =

The following are some properties of these functions:

ℎ −ℎ  −ℎ  ℎ  ℎ − ℎ  ℎ− −ℎ ℎ− ℎ ℎ − −ℎ ℎ   ℎℎ ℎℎ ℎ    ℎℎ ℎℎ  ℎ ℎ    ℎ ℎ ℎ 2

2

2

1

2

=1 2

= 1=

2

= = =

( ± )= ±

=

±

=

±

±

±

1±

24

1

2

ℎ    − − ℎ =  −− ℎ  =    − ℎ   − 1

=

2

+

+

st




Inverse Hyperbolic Functions

When hyperbolic function keys are not available on a calculator, it is still possible to evaluate the inverse hyperbolic functions by expressing them as logarithms, as shown below: (a) (b) (c) (d) (e) (f)

ℎ−  ℎ−  ℎ−  ℎ−  ℎ−  ℎ− 

         −    −        − 

1

= ln

+

2

+1 ,

1

= ln

+

2

+1 ,

1

=

1

1 2

= ln

1

= ln

1

=

1 2

1+

ln

,

1

2

1+ 1

1

+

ln

1+ 2

+ 1 1

,

,

,

 ≥   ≤ ≠  1

<1

0<

1

0

>1

Example (12): Identify each function given here as one of the types of functions we have discussed. Keep in mind that some functions can fall into more than one category. For example, = 2 is both a power function and a polynomial of second degree.

 

25

st


Shifting a Graph of a Function 



 

   −     −        

To shift the graph of a function = ( ) straight up, add a positive constant to the right hand side of = . To shift the graph of a function = ( ) straight down, add a negative constant to the right hand side of = ( ). To shift the graph of = ( ) to the left, add a positive constant to x. To shift the graph of = ( ) to the right, add a negative constant to x.

Shift Formulas Vertical Shifts 

   =



+

Shifts the graph of f up Shifts it down

Horizontal Shifts

  ℎ =

+

 



units if k > 0

ℎ

units if h > 0

units if k < 0

Shifts the graph of f left Shifts it right

ℎ 

units if h < 0

EXAMPLE (13): Shifting a Graph (a) Adding 1 to the right-hand side of the formula the graph up 1 unit (Fig.1.27).

−

       − =

(b) Adding 2 to the right-hand side of the formula shifts the graph down 2 units (Fig.1.27).

 

2

to get

=

2

=

to get

2

+ 1 shifts =

Fig.(1.27): To shift = 2 up (or down), we add positive (or negative) constants to the formula for f (Example 13a and b). 26

2

2

st


(c) Adding 3 to x in (Fig.1.28).

  =

2

to get

 

= ( + 3)2 shifts the graph 3 units to the left

 

Fig.(1.28): To shift the graph = 2 to the left, we add a positive constant to . To shift the graph to the right, we add a negative constant to (Example 13c).

− −  −

  



−



(d) Adding 2 to x in = and then adding 1 to the result, gives = 2 1 and shifts the graph 2 units to the right and 1 unit down (Fig.1.29).

 

Fig.(1.29): Shifting the graph = 2 units to the right and 1 unit down (Example 13d).

27

st


Homework (2):

28

st


29

st


30

st


31

st


32

st


33

st


34

st


35

st


36

Ch1 - Algebraic and Non-Algebraic Functions and Their Inverse

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