MAT121: SECTION 2.7 ANALYZING GRAPHS AND PIECEWISE FUNCTIONS SYMMETRY, EVEN, ODD A graph can be symmetric about the x-axis, y-axis, or the origin (y = x). If a mirror is placed on those lines, the graph would reflect itself. SYMMETRY ABOUT X-AXIS A graph that is symmetric about the x-axis satisfies the property y = y. Note: Such a graph is not a function. It fails the VLT. Section 2.7 Analyzing Graphs and Piecewise Functions 1
SYMMETRY ABOUT THE Y-AXIS A graph that is symmetric about the y-axis satisfies the property f( x) = f(x). It is an EVEN function. The simplest example is: f(x) = x 2 NOTE: f( x) = f(x) f(2) = (2) 2 = 4 f( 2) = ( 2) 2 = 4 SYMMETRY ABOUT THE ORIGIN A graph that is symmetric about the origin satisfies the property f( x) = f(x). It is an ODD function. The simplest example is: f(x) = x 3 NOTE: f( x) = f(x) f( 1) = ( 1) 3 = 1 f(1) = (1) 3 = 1 Section 2.7 Analyzing Graphs and Piecewise Functions 2
IDENTIFYING EVEN, ODD, NEITHER Consider m(x) = x 5 + x 3. Is the function even, odd, or neither? Test for ODD: m( x) = m(x) m( x) = ( x) 5 + ( x) 3 = x 5 x 3 m(x) = ( x 5 + x 3 ) = x 5 x 3 The function m(x) passes the test for ODD. Consider n(x) = x 2 x + 1. Is the function even, odd, or neither? Test for EVEN: n(x) = n( x) n( x) = ( x) 2 + x + 1 = x 2 x + 1 The function n(x) passes the test for EVEN. Consider p(x) = 2 x + x. Is the function even, odd, or neither? Test for EVEN: p(x) = p( x) p( x) = 2 x + ( x) = 2 x x The function p(x) fails the test for EVEN. Test for ODD: p( x) = p(x) p(x) = ( 2 x + x) = 2 x x The function p(x) fails the test for ODD. Therefore, the function p(x) is NEITHER. Section 2.7 Analyzing Graphs and Piecewise Functions 3
ON YOUR OWN Determine even, odd, or neither. 1. Q(x) = 16 + x 2 Q( x) = 16 + ( x) 2 = 16 + x 2 Q(x) = Q( x) so EVEN 2. r(x) = 4x 2 + 2x 3 r( x) = 4( x) 2 + 2( x) 3 = 4x 2 2x 3 r(x) r( x), so NOT EVEN r(x) = (4x 2 + 2x 3) = 4x 2 2x + 3 r( x) r(x), so NOT ODD NEITHER PIECEWISE FUNCTIONS There s a good real-world example on page 279. Students are left to read it on their own. Consider Let s look at a number line. (Created during lecture.) Evaluate the following. f( 3) If x = 3, then use the first expression, x + 7. f( 3) = 4 f ( 2 3 ) If x = 2 3, then use the second expression, x2. f ( 2 3 ) = 4 9 f(2) If x = 2, then use the third expression, 3. f(2) = 3 Section 2.7 Analyzing Graphs and Piecewise Functions 4
ON YOUR OWN Evaluate f( 2) and f(5). f( 2) = ( 2) 2 = 4 f(5) = undefined GRAPHING PIECEWISE FUNCTIONS Consider What is the domain for f(x)? [ 4, ) (Graphed by hand during lecture.) Graphing by Calculator o Y1= (abs(x))(x 4)(x < 2) o Y2= ( x + 2)(x 2) o ABS is under MATH > NUM> 1: o Inequality Symbols are under TEST (2 nd MATH) ON YOUR OWN Graph the following piecewise. Section 2.7 Analyzing Graphs and Piecewise Functions 5
INCREASING, DECREASING, CONSTANT BEHAVIOR Read all graphs from left to right. Use interval notation and always round brackets ( ) when stating intervals of increasing, decreasing, and constant behavior. The endpoints are neither increasing, decreasing, nor constant. The intervals are expressed using the values of the DOMAIN. Increasing: (2, ) Decreasing: ( 3, 2) Constant: ( 2, 2) Consider the following functions. Determine intervals of increasing, decreasing, and constant. Increasing: (, ) Decreasing: (, 0) (2, ) Constant: (0, 2) Section 2.7 Analyzing Graphs and Piecewise Functions 6
RELATIVE (LOCAL) MINIMA AND MAXIMA Formal Definition is on page 285. The x-coordinate is the location of a max/min. The y-coordinate is the value of a max/min. There can be multiple maximums, multiple minimums, only one (x 2 ), or none (x 3 ). For HW/Test, include both value and location. The relative maximum of f(x) is 0 and is located at x = 0. The relative minimum of f(x) is 4 and is located at x = ±2. Max: f(0) = 0 Min: f(±2) = 4 What are the intervals of increasing/decreasing? Increasing: ( 2, 0) (2, ) Decreasing: (, 2) (0, 2) What is the relationship between these intervals and the max/min? The endpoints of the intervals are the locations of the maximums and minimums. USING THE CALCULATOR TO FIND MAX/MIN 1. Define your function in Y1=. 2. CALC 3: minimum OR 4: maximum 3. Left Bound; Right Bound; Guess Practice: x 3 4x 2 + 3x PARTING THOUGHT: Suppose that the average rate of change of a continuous function between any two points to the left of x = a is positive, and the average rate of change of the function between any two points to the right of x = a is negative. Does the function have a relative minimum or maximum at a? The function has a maximum at a. Section 2.7 Analyzing Graphs and Piecewise Functions 7