Ch. 8.7 Graphs of Rational Functions Learning Intentions: Identify characteristics of the graph of a rational function from its equation. Learn to write the equation of a rational function from its graph. Rewrite a function as a rational function. Review factoring. p( x) f ( x) = k + a q( x) Identify holes, vertical & horizontal asymptotes and/or x & y-intercepts of rational functions. x h
Types of Asymptotes
Rational Function Patterns
p.500 Ex.B.) Describe the features of the graph y = x2 + 2x 3 x 2 2x 8 Features (i.e. holes, asymptotes & intercepts) of these functions are apparent when the numerator & denominator are factored. Holes Graph Features Location Equation feature x-intercepts Vertical asymptotes y-intercepts Horizontal asymptotes Visit the Desmos site below to help determine the relationship between rational functions & their graphs. https://www.desmos.com/calculator/ng71ls9re2
SOLUTIONS: p.500 Ex.B.) Describe the features of the graph y = x2 + 2x 3 x 2 2x 8 Features of these functions are apparent when the numerator & denominator are factored. Thus, y = x2 + 2x 3 x 2 2x 8 = (x + 3)(x 1) (x 4)(x + 2)
Ch. 8.7 Investigation Goal: Identify characteristics (holes, asymptotes, x & y-intercepts) of graphed rational functions from their equations. Step 1: i.) Match each rational function with a graph. ii.) Investigate each graph by dragging a point on the graph, tracing or looking at a table of values. Explain which method you used & why. ii.) What features in the equation cause the different types of graph behaviors?
Step 2: Ch. 8.7 Investigation i.) Find a rational function equation for each graph. ii.) Write a few sentences that explain the appearance of your graph & why your equation creates these particular outcomes & behaviors. Step 3: Write a paragraph or a set of guidelines that explain how you can use: i.) an equation to predict the graph - particularly, where holes & asymptotes will occur. ii.) features of a graph to write an equation. Step 4: Consider the graph of: y = x 2 (x 2) 2 i.) What features does this graph have? ii.) What can you generalize about the graph of a function that has a factor that occurs more times in the denominator than in the numerator?
Step 1 Ch. 8.7 Investigation KEY 1a.) = B; vertical asymptote x = 2. x = 2 makes the denominator 0 and this is a translation of the parent hyperbola 1 x horizontally 2 units (h = 2). 1b.) = D; hole at x = 2. x = 2 makes both the numerator & denominator 0 and y reduces to 1 for all x= 2. 1c.) = A; vertical asymptote x = 2. x = 2 makes the denominator 0, and the denominator is squared so that all values become positive. 1d.) = C; hole at x = 2. x = 2 makes both the numerator and denominator 0. If you reduce the equation, you get y = x 2, which produces the graph shown, except you get a hole at x = 2.
Step 2a.) y = 1 x+1 Ch. 8.7 Investigation KEY. The graphic has asymptote x = -1. Other than that, the equation looks like y = 1 x. Step 2b.) y = 2(x+ 1) x+1. There is a hole at x = -1, so the denominator is x + 1. However, when reduced, it equals the line y = 2, so there must be factors of 2 & an x + 1 in the numerator. 1 Step 2c.) y = (x+1) 2. There is a vertical asymptote at x = -1, so the denominator includes x + 1. Because all y-values end up positive, the equation must have an even number of factors of x + 1 in the denominator. Step 2d.) y = (x+1)2 x+1. There is a hole at x = -1, so the denominator includes x + 1. When the equation is reduced, it must be y = x + 1to produce the given graph, so the equation must be y = (x+1)2 x+1.
Step 3 A vertical asymptote occurs at zeros that appear only in the denominator or at zeros that appear more times in the denominator than in the numerator. A hole occurs at values that make both the denominator and the numerator 0, provided there are no vertical asymptotes at these values. Step 4 The graph has vertical asymptote x = 2. When a factor occurs in both the denominator & the numerator, but occurs more times in the denominator, it indicates a vertical asymptote rather than a hole. NOTE: You might think that because the value x 2 makes both the numerator and denominator equal 0, it indicates a hole. When you reduce a rational expression, like y = y = 1 x 2 x 2 (x 2) 2 to, you may lose information about a value for which the denominator is undefined. In this case, however, the factor (x 2) is still represented in the denominator, so y = x 2 (x 2) 2 is equivalent to y = 1 x 2. This function has a vertical asymptote, not a hole.