Analyzing and sketching the graph of a rational function Rational Functions

Rational Functions? A function of the form f ( x ) = N ( x )/ D ( x ), where N and D are both polynomials. Examples:

Characteristics we will find for every problem Found with algebra 1. Horizontal Asymptote (HA) (we’ll see how to find in a minute) 2. Vertical Asymptote (we’ll see how to find in a minute 3. Holes (we’ll see how to find in a minute) 4. Points of Discontinuity – VA and holes 5. y- intercept (set x=0) 6. x- intercept (set numerator = 0) 7. Rate of change - find y values for 2 given x values, then use slope formula to calculate

Characteristics found on graph 8. Domain – all x values, VA and holes will be excluded 9. Range – all y values 10. Extrema – y values that are a minimum or maximum – they can be relative or absolute 11. End behavior – what happens to the y-values on left and right sides, will match the HA 12. Intervals of Increase (II)– where graph is going up, we’ll find on the graph 13. Intervals of Increase (ID)– where graph is going down, we’ll find on the graph

A function is CONTINUOUS if you can draw the graph without lifting your pencil. A POINT OF DISCONTINUITY occurs when there is a break in the graph. Continuity There are 2 types of discontinuity we will look at: Asymptotes Holes

Asymptote: a line that the graph approaches more and more closely but will never touch. 1. vertical asymptotes: what ____ CANNOT be 2. horizontal asymptotes: what ____ CANNOT be DISCONTINUITIES Hole: a single point at which the graph has no value

Horizontal Asymptotes Compare the largest exponents in numerator and denominator BOBO BOTN EATS D/C Bigger On Bottom y=0 Bigger On Top – None Exponents Are The Same; Divide the Coefficients

FINDING HA If the bigger exponent is on the bottom y = 0 Examples:

FINDING HA If the exponents are equal, divide their coefficients Examples:

SLANT ASYMPTOTES If the higher exponent is on top, there is a SLANT asymptote. We will not learn slant asymptotes so we will write “NONE” Examples:

HOLES VS VA A hole is a factor which reduces with a factor in the numerator. A VA is a factor which doesn’t reduce with a factor in the numerator. Examples:

YOU TRY Find the x-intercepts, HA VA and Hole

FINDING THE X-INTERCEPTS set the numerator = to 0 and solve for x the x-values are the x-int. Unless the HA is y=0, then NONE x-intercepts:

Find the x-intercepts, y-intercepts, VA, holes and HA: x-intercepts: y-intercepts: VA: Holes: HA: x-intercepts: y-intercepts: VA: Holes: HA: x-intercepts: y-intercepts: VA: Holes: HA: x-intercepts: y-intercepts: VA: Holes: HA:

Rate of Change 2 x values will be given to you Plug them in to the equation and find the y values Write these as 2 points Use the slope formula to calculate the rate of change

Find the Rate of Change between x=3 and x=5

Domain is all the x-values, VA and holes will be excluded. So it will be found the same as VA and holes, just written In a different form. DOMAIN

Determine the domain of these rational functions:

End Behavior To find the end behavior, To find the end behavior, look at the HA. The end behavior will look at the HA. The end behavior will be equal to the HA and will be written in limit notation

On the right side, the x values keep Getting larger and larger and approach infinity, the y values get close to 0 What if x is a negative number? It gets closer to infinity, the y value gets close to 0 End Behavior

GRAPHING RATIONAL FUNCTIONS Steps to graphing rational functions: 1.find the horizontal asymptote 2.Find the vertical asymptotes and holes 3.Find the x-intercept 4.Find the y-intercept (plug in 0 for x) 5. Graph 6. find the characteristics

Graph of a Rational Funtion HA: y=0 VA: x=1 Hole: x=-1 x- int: none y- int: y=-1

Characteristics 1. HA 2. VA 3. Holes 4. y- intercept 5. x- intercept 6. Points of Discontinuity 7. Rate of change between x = 2 and 4 8. Extrema 9. Domain 10. Range 11. End behavior 12. II 13. ID

Graph: 1 st, find the vertical asymptote. 2 nd, find the x-intercept. 3 rd, find the y-intercept. 4 th, find the horizontal asymptote. 5 th, sketch the graph.

Characteristics 1. HA 2. VA 3. Holes 4. y- intercept 5. x- intercept 6. Points of Discontinuity 7. Rate of change between x = 2 and 4 8. Extrema 9. Domain 10. Range 11. End behavior 12. II 13. ID

Graph: 1 st, factor the entire equation: 2 nd, find the x-intercepts: 3 rd, find the y-intercept: 4 th, find the horizontal asymptote: 5 th, sketch the graph. Then find the vertical asymptotes:

Characteristics 1. HA 2. VA 3. Holes 4. y- intercept 5. x- intercept 6. Points of Discontinuity 7. Rate of change between x = -1 and 1 8. Extrema 9. Domain 10. Range 11. End behavior 12. II 13. ID

Example

Characteristics 1. HA 2. VA 3. Holes 4. y- intercept 5. x- intercept 6. Points of Discontinuity 7. Rate of change between x = -1 and 1 8. Extrema 9. Domain 10. Range 11. End behavior 12. II 13. ID

Example

Characteristics 1. HA 2. VA 3. Holes 4. y- intercept 5. x- intercept 6. Points of Discontinuity 7. Rate of change between x = 3 and 6 8. Extrema 9. Domain 10. Range 11. End behavior 12. II 13. ID