5.1N Key Features of Rational Functions

Transcription

1 5.1N Key Features of Rational Functions A. Vocabulary Review Domain: Range: x-intercept: y-intercept: Increasing: Decreasing: Constant: Positive: Negative: Maximum: Minimum: Symmetry: End Behavior/Limits: x B. Asymptotes An Asymptote 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: 1. Identify 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.

2 C. End Behavior and asymptotes 1. After you read through and discuss the following information, complete 5.1A. 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. x What value is the graph approaching on the negative(left) side of the graph? What value is the graph approaching on the positive(right) side of the graph? Hint: When the limit shows or look at the horizontal asymptote.

3 D. Identify key features of a rational function. Example 1 Example Domain: Positive: Range: Negative: x-intercept(s): Maximums / minimums: y-intercept: Symmetry: Increasing: End Behavior/Limits: Decreasing: Constant: Vertical Asymptote(s): Horizontal Asymptote: x Example Domain: Positive: Range: Negative: x-intercept(s): Maximums / minimums: y-intercept: Symmetry: Increasing: End Behavior/Limits: Decreasing: Constant: Vertical Asymptote(s): Horizontal Asymptote: x

4 E. Finding x and y intercepts. Remember when finding the x-intercept, the y-value is 0; (x, 0). To use an equation to find the x-intercept, substitute 0 in for y and solve for x. Remember when finding the y-intercept, the x-value is 0; (0, y). To use an equation to find the y-intercept, 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) x-intercept x-intercept x-intercept y-intercept y-intercept y-intercept 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 f( x) x 49 a) f(4) a) x 1, y a) f(0) b) f(3) b) x 1, y b) f() 3

5 G. Sign Arrays. There are times when we need to know where a graph is positive (above the x-axis) and negative (below the x-axis) 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 x-intercepts are what we use to divide a graph into its positive and negative intervals, we need to find the x-intercepts 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 x-intercepts. Factor when necessary. Step : Plot the x-intercepts on a number line that represents the x-axis. Step 3: Pick any point that lies in each of the intervals or sections of the graph created by the x-intercepts. Substitute that number into the equation and evaluate. Step 4: If the answer is positive, that means the graph is positive in that section. If 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: If 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 x-intercepts as the graph. The intervals on the sign array marked positive should match where the graph is above the x-axis and the intervals on the sign array marked negative should match where the graph is below the x-axis. Positive Above x-axis Negative Below x-axis

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) xx ( 6)( x 7) x-intercepts: x-intercepts: