Section 2.5: Graphs of Functions

Transcription

1 Section.5: Graphs of Functions Objectives Upon completion of this lesson, ou will be able to: Sketch the graph of a piecewise function containing an of the librar functions. o Polnomial functions of degree less than 4 Constant Linear Quadratic Cubic o Absolute value functions o Rational functions o Square root functions Answer questions about functions of the form: o as a, f ( ), or as a +, f ( ), or as a, f ( ), and o f (a) =. Given a graph or equation of a function, sketch its absolute value. Given a piecewise verbal model, write a mathematical model. Required Reading Swokowski/Cole: Section.5, pages Focus on eample 8 on page 16 and eample 1 on page 167. Discussion Sections.4 and.5 use man basic functions in the eamples. Below is a summar of the basic functions ou will want to familiarize ourself with. Librar of Functions n n 1 n Polnomial Functions are of the form: f ( ) = a + a + a a + a + a The domain of all polnomial functions is the set of real numbers. We will limit our discussion of polnomials to linear, quadratic, and cubic functions. n n 1 n 1 0 Linear functions are of the form: f ( ) = a1 + a0 or = m + b Special cases of liner functions include: Identit Function: =

2 = This function has a slope of 1 and a -intercept of 0. The identit function is an odd function; it is smmetric with respect to the origin. Constant Function: = b = b This function has a slope of 0 and a -intercept of b. It is a horizontal line through b. An constant function is an even function; it is smmetric with respect to the -ais. Quadratic functions are of the form: f ( ) = a + a + a 1 0 or = a + b + c b The graph of a quadratic function is a parabola with a verte at a f b,. a b The ais of smmetr of the parabola is the vertical line = a. The parabola opens up if a > 0 and down if a < 0. The simplest quadratic function is the square function: = The graph of the square function has its verte at the origin. It is an even function.

3 3 Cubic functions are of the form: f ( ) = a + a + a + a or = 3 a + b + c + d The simplest cubic function is the equation: = 3 The graph of this simple cubic function is odd. Square Root Function The simplest square root function is the equation: = Rational Function The simplest rational function is the reciprocal function: = 1 This is an odd function with a vertical asmptote along the -ais. Absolute Value Function The simplest absolute value function is: B definition, if 0 = if < 0 Therefore, the graph is = if 0 and = - if < 0. This simple absolute value function is even. = 3

4 Greatest-Integer Functions Even though there are man applications of the greatest-integer function, ou are not responsible for them. Piecewise Functions Piecewise functions will consist of pieces of the librar functions mentioned above, or slight variations to their simplest forms, defined for a limited domain. Following is an eample of a piecewise function. This particular eample is composed of three of the basic functions mentioned above. if 0 Eample 1: Sketch the graph of the piecewise function f ( ) = 5 if 0 < <. 3 if Solution: The first part of this function, f ( ) =, is the square root function defined for non-positive real numbers. The second part of this function, f ( ) = 5, is a constant function defined for all real numbers between 0 to. The third part of this function, f ( ) = 3, is a linear function defined for real numbers of and greater. 4

5 From the sketch and the given function, we can see that: a. f ( 3) = 3 b. f ( 15. ) = 5 c. f ( 3) = 9 d. f ( 0) = 0 e. As 0, f ( ) 0. This notation means that as approaches 0 from the left, f() is approaching 0. + f. As 0, f ( ) 5. This notation means that as is approaching 0 from the right, f() is approaching -5. g. f ( ) = 6. h. As, f ( ) 5. + i. As, f ( ) 6. The following is an eample of an absolute value function. Eample : Sketch the graph of the absolute value function f ( ) = + 3. Solution: Think of the function f ( ) as f ( ) = g( ) where g( ) = + 3. The function g() is an absolute value function shifted left units and down 3 units as shown below. g() The graph of f() is just a modification to the graph of g() in which the negative values are converted to their positive value. See below. 5

6 f() Practice Problems Work these problems. Answers to the odd numbered problems can be found at the end of our tet, even answers are below. Section Pages Eercises , 49, 50, 51, 5, 57, 58, 59, 60, 61, 6, 65, 67 Answers to even eercises

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