1 GRAPHING RATIONAL FUNCTIONS DAY 2 & 3 Unit 12
2 Warm up! Analyze the graph Domain: Range: Even/Odd Symmetry: End behavior: Increasing: Decreasing: Intercepts: Vertical Asymptotes: Horizontal Asymptotes: Domain:, 2 2,2 2,4 4, Range:,0 0, Symmetry: none End behavior: x -, f(x) 1 x +, f(x) 1 Intercepts: x = - 3, 3, 5 and y=3 Vertical Asymptotes: x = 4, x = 2, x = - 2 Horizontal Asymptotes: y = 1
3 Holes What are the yellow circles? They are called holes.
4 Holes & Vertical Asymptotes Investigation What causes a vertical asymptote? When the denominator equals zero. What causes a hole in the graph? Holes occur in a rational function when the same binomial, (x - a) for example, exists in both the numerator and denominator. How can you tell where the vertical asymptote(s) will be? Set the denominator to zero and solve for x. How can you tell where the holes will be? Holes are located at x = a.
5 Asymptotes Do you notice something different between horizontal asymptotes and vertical asymptotes? The graph crosses over the horizontal asymptote at x=3 Not all graphs of rational functions have horizontal asymptotes. From the last class horizontal asymptotes occur when nd < dd y = the value f(x) as x - and x + nd = dd y = leading coefficient of numerator/leading coefficient of denominator What if nd > dd?
6 Asymptotes Investigation What causes a horizontal asymptote? What causes a slant asymptote? How can you tell where a horizontal asymptote will be? How can you find the equation of the line for a slant asymptote? When the degree of the numerator is less than or equal to the degree of the denominator When the degree of the numerator is greater than the degree of the denominator If dn<dd then y=0 and if dn=dd then y=ratio of leading coefficients Use long division. The equation is the quotient (neglect the remainder).
7 Warm up! Graph = 2 **remember to rewrite the function with a common denominator!** Is f(x) a transform of the parent function 1/x?? Y-intercept: (0,1) X-intercept: (0.5,0) Vertical asymptotes: x=-1 Horizontal asymptotes: y=-2 No holes
8 Equation of a Slant Asymptote To find the equation of a slant asymptote, we divide the rational using long division. The equation is the quotient (neglect the remainder). Example: = 2 2 3 3 +3
9 Putting It All Together: Example 1 Graph = Step 1: Find the y-intercept Evaluate f(0) 0 = No y-intercept Step 2: Find the x-intercepts Set the numerator to zero and solve for x +2=0 x=-2 (-2,0) Step 3: Find the vertical asymptote(s) Factor the denominator = Check for holes Common factor of (x+2) x 2
10 Putting It All Together: Example 1 Graph = Simplify the function = Set the denominator to 0 4x=0 Vertical asymptote at x=0 Step 4: Find horizontal asymptotes Compare the degree of the numerator (nd) and denominator (dd) nd = 1 < 2 = dd Horizontal asymptote at y=0 Step 5: Plot intercepts, hole(s) and asymptote(s) Step 6: Find at least one point between each x-intercept and vertical asymptote
11 Example 1 (cont.) = 0, 2 x -2 x=0 y=0 x y -3 -.08-1 -.25 -.5 -.5.5.5 1.25 2.125
Example 1 (cont.) 12
13 You Try! Graph = Step 1: (0,0) Step 2: (0,0) multiplicity of 3 Step 3: (x-2)(x+2) no holes x=-2, x=2 Step 4: 3 < 2 y=x Step 6: x y -5-5.95-1 0.33 1-0.33 5 5.95
14 Writing Equations Given Key Features Example 1 Write a rational function that has known zeros at (2,0) and (-3,0), a horizontal asymptote at y=2, and a vertical asymptotes at x=±1 Known zeros tell us that the numerator includes (x-2)(x+3) Horizontal asymptote at y=2 tells us that nd=dd and that the ratio of the leading coefficients is 2/1 Vertical asymptotes tell us the denominator includes (x-1)(x+1) =
Writing Equations Given Key Features Example 2 Write a rational function that has a known zero at (0,0), a slant asymptote at y = x + 1, a hole at (2,4), and a vertical asymptote at x=1. Known zero tell us that the numerator includes (x-0) Hole tells us x 2 and the numerator and denominator includes (x-2) Vertical asymptotes tell us the denominator includes (x-1) So far we have = 15
Writing Equations Given Key Features Example 2 (cont.) So far we have = Slant asymptote tells us that nd>dd. Which term in the numerator should we square? Let s try squaring the first term first First foil/distribute the function = Using long division, the quotient should be the equation of the slant +1 3 +2 2 +3 2 It s a match! = 2 1 2 16
17 Writing Equations Given a Graph Example 1 X-intercept: (2,0) Hole: x 5 Vertical Asymptote: x=-1 Horizontal Asymptote: y=2 =
18 Writing Equations Given a Graph Example 2 X-intercept: (-2,0) Hole: x 3 Vertical Asymptote: none Horizontal Asymptote: none = =
19 You Try! X-intercept: none Hole: x -5 Vertical Asymptote: x=-3 Horizontal Asymptote: y=0 =