WARM UP: Factor each completely b) c) a) d) e) f)

Ratio or Fraction of TWO polynomials Rational Expression/ Equations: Ratio or Fraction of TWO polynomials Previous Knowledge: (1) SIMPLIFYING (x + 5)(x – 3) a) b) (x + 5)(x + 3) (2x - 5)(x + 2) c) b) a2(b – 1) a2(b – 1) a3(1 – b) -a3(b –1) (x + 2)(x + 8)

Previous Knowledge: (2) MULTIPLYING and DIVIDING (x + 4)(x – 2) 4(x + 1) a) b) (x + 6)(x + 1) c) d)

Previous Knowledge: (3) ADDITION and SUBTRACTION

EVALUATING RATIONAL FUNCTIONS Substitute x-value into the function Evaluate the numerator and denominator separately Reduce the resulting fraction

Practice: Evaluating Rational Functions b)

Practice: Continued c) d)

Number Lines and Rational Equations 1) Find All Zero/ Root Values of the numerator and denominator ZEROS in the numerator = zero in the graph (y-value) ZEROS in the denominator = undefined in the graph (no y-value) 2) Write all the Zero/Undefined x-values on the number line 3) Between those x-values pick a number to substitute into function to determine if the graph (y-value) will be positive or negative REASON: Only time we can change from positive to negative value is when we have a ZERO or UNDEFINED value of the graph Example:

Number Lines and Rational Equations [1] [2]

Number Lines and Rational Equations [3[ [4]

Unit 4: Graphing Rational Equations Graph each of the following rational equations in your calculator and Sketch the graph on the provided axes [1] [2] [3] [4] [5] [6]

Observations based on the graphs #1 – 6: Investigation: Graphing Rational Equations Observations based on the graphs #1 – 6: (1) What is different about these graphs from previous functions that you have drawn? Separate sections of the graph (discontinuity) Approaching specific lines (asymptotes) Holes in some graphs All decreasing, All Increasing, or Valley/ Hill Sections (2) Is there any relationship you see between the numerator and/or denominator with the behavior of the graph? Denominator Values Affect the vertical lines Numerator/Denominator combined affect the horizontal line

Graphing Rational Equations By Hand Basic Steps 1. Factor Numerator and Denominator 2. Determine ZERO(S) of Denominator and Numerator 3. Determine the types of Asymptotes and Discontinuity (Use zeros to help) 4. DRAW Asymptotes 5. GRAPH based on known values or positive/negative sections

Special Behavior in Rational Equations #1: Vertical Asymptotes x = a is a vertical asymptote if f(a) is undefined and a is a zero value of the denominator of f(x) only. As x approaches a from the left or right side, f(x) approaches either ±∞ “Boundary you follow along” #1: x = a #2: x = a #3: x = a #4: x = a Examples: Zeros of Denominator that do not cancel Vertical Asymptotes at x = 1 and x = -6

Special Behavior in Rational Equations #2: Points of Discontinuity (Holes in Graph) x = a is a point of discontinuity if f(a) is undefined a is a zero value of the numerator and denominator of f(x). Factor (x – a) can be reduced completely from f(x) #4: x = a #1: x = a #2: x = a #3: x = a Example: Zeros of Denominator that cancel Point of Discontinuity at x = -3 Vertical Asymptote at x = 5

Special Behavior in Rational Equations #3 Horizontal Asymptotes: y = b is a horizontal if the end behavior of f(x) as x approaches positive or negative infinity is b. Note: f(x) = b on a specific domain, but is predicted not approach farther left and farther right Case 1: Degree of denominator is LARGER than degree of numerator Horizontal Asymptote: y = 0 (x – axis) Case 2: Degree of denominator is SAME AS degree of numerator Horizontal Asymptote: y = fraction of LEADING coefficients Case 3: Degree of denominator is SMALLER than degree of numerator No Horizontal Asymptote: f(x) → ± ∞

Example 1: Sketch two possible graphs based on each description Vertical: x = 2 Horizontal: y = - 3 Discontinuity: x = - 4 [2] Vertical: x = -4, x = 0 Horizontal: y = 0 Discontinuity: x = 2

Example 2: Determine the asymptotes and discontinuity values for the given rational equation and plot them on the given axes FACTOR NUMERATOR AND DENOMINATOR!!!! (x + 2)(x + 3) a) (x + 2)(x – 5) pd VA: HA: PD: x = 5 y = 1 x = – 2

Example 2 Continued (x + 2)(x – 4) b) (x + 3)(x – 4) pd VA: HA: PD: x = – 3 y = 1 x = 4

Example 2 Continued c) VA: HA: PD: x = 4 y = 2 NA

Example 3: Determine the asymptotes and discontinuity values for the given rational equation b) VA: HA: PD: x = 6 x = 6 VA: HA: PD: x = 0 y = 5/2 y = 0 x = 3 NA

Example 3 Continued c) d) (x + 2)(x – 5) VA: HA: PD: x = 5, x = – 2 VA: HA: PD: x = 1/4, x = 3/5 y = 0 y = 3/10 NA x = -2

PRACTICE: Identify the vertical asymptotes, horizontal asymptotes, and points of discontinuity if they exist in each rational equation 1. 2. 3. 4.

PRACTICE: continued 5. 6. 7. 8.