obj Domain: Elis ] Range: x-intercept: ( O - y-intercept: 1-4 Increasing: Constant: NA HA NA

Transcription

1 1/6/18 51N Key Features of Rational Functions Vocabulary Review obj : omain: Positive: G 4 ) Range: Negative: Elis ] f hind xintercept: Maximum: 3 C 40 ) ( no ) O yintercept: Minimum: ( O ncreasing: Symmetry: no ) End Behavior/Limits: ecreasing: tho )vlh3 ) Constant: N H N B symptotes n symptote is a line that a curve approaches as it heads towards infinity or negative infinity There are three types of asymptotes: horizontal vertical and oblique Examples: wwwmathisfuncom/algebra/asymptotehtml 1 dentify the asymptote(s) in each graph below and state whether it is horizontal vertical or oblique Write an equation that represents each asymptote and label it on each graph a b c V x V X x3 V X 4 H 90 H y 1 H Y

2 C End Behavior and asymptotes 1 fter you read through and discuss the following information complete 51 x The graph is approaching the asymptote x from the positive(right) side x The graph is approaching the asymptote x from the negative(right) side This is the vertical asymptote x This is the vertical asymptote x What value is the graph approaching on the negative(left) side of the graph? Hint: When the limit shows or look at the horizontal asymptote What value is the graph approaching on the positive(right) side of the graph?

3 dentify key features of a rational function Example 1 Example omain: ) Range: Positive: Negative: xintercept(s): ) Maximums / minimums: yintercept: VloNU/4Hollho N/ None Symmetry: N/ ncreasing: End Behavior/Limits: ecreasing: Constant: C and to Vertical symptote(s): Horizontal symptote: x 0 x 4 ( 10)U( 14) to o)ul0m)vh 3 3 N/ N/ XOx4 y 3 x 0 x 4 a Example omain: Positive: too caswtsivuio ) 51040) Range: Negative: to Uco C a ) s ) xintercept(s): Maximums / minimums: N 0 yintercept: %) Symmetry: ncreasing: End Behavior/Limits: ecreasing: Constant: Vertical symptote(s): Horizontal symptote: 40 ( O to guts N/ ) x 0 5 x a4 x 0 st x 4 Yz CVKio) O O N µ

4 (311313)4 n gt E Finding x and y intercepts Remember when finding the xintercept the yvalue is 0; (x 0) To use an equation to find the xintercept substitute 0 in for y and solve for x Remember when finding the yintercept the xvalue is 0; (0 y) To use an equation to find the yintercept substitute 0 in for x and solve for y Example 1: Example : Example 3: y 1 x1 x 4 3x y 8 f(x) (x3)(x4) y int tint 5 tent ' xintercept xintercept xintercept yintercept 1041 yintercept to yintercept " " " " 0 1 y 's T H CEO) x tho ) 4 ) l O y f 's F Evaluate Functions Evaluate the following functions and equations with the given values Substitute the given value in for x and solve for y 11 f( x) x 3x 4 1 y 6x 7x3 x 9 x 5 13 f( x) x 49 3 T a) f(4) a) x 1 y a) f(0) ta to 4 O 049 b) f(3) b) x 1 y b) f() ) } Ea ' s 45

5 5) G Sign rrays There are times when we need to know where a graph is positive (above the xaxis) and negative (below the xaxis) without graphing the function We do this by making a sign array Big Question: Make a sign array of the following equation to find where the graph is positive and negative f x x x ( ) 4 5 Since the xintercepts are what we use to divide a graph into its positive and negative intervals we need to find the xintercepts first Remember to do this in a quadratic function means you have to factor and set each factor equal to zero Then solve for x Step 1: Find the xintercepts Factor when necessary xh 4 5 o L x Htt ) S Step : Plot the xintercepts on a number line that represents the xaxis Step 3: Pick any point that lies in each of the intervals or sections of the graph created by the xintercepts Substitute that number into the equation and evaluate t 1714 ) S ) S Step 4: f the answer is positive that means the graph is positive in that section f the answer is negative that means the graph is negative in that section Mark each section on the number line either positive() or negative() Summary/Making connections: f you compare the graph of the function f( x) x 4x5 to the sign array you will see that the sign array has the same xintercepts as the graph The intervals on the sign array marked positive should match where the graph is above the xaxis and the intervals on the sign array marked negative should match where the graph is below the xaxis Positive bove xaxis Negative Below xaxis

6 Examples: Make a sign array for each equation to find where the graph is positive and negative 1 f( x) x x 0 f( x) x( x 6)( x 7) factor 01 5)LXt4 ) S 4 to find zeros X XO 60 xt70 xintercepts: 40 ) 40) xintercepts: ( ) t mmmm est )t8t >(67)7 So graph goes iit#u :#