An asymptote is a line that a curve approaches on a graph.
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1 An asymptote is a line that a curve approaches on a graph. Vertical asymptote at x = -3 Horizontal asymptote at y = 1
2 Oblique (slanted) asymptote at y = -.5x -.5
3 Vertical Asymptotes Vertical asymptotes occur when the denominator of a rational function equals zero. The domain of a rational function is all x-values except where there are vertical asymptotes or the function is undefined. Ex) Find the vertical asymptotes and domain of: f(x) = 3 x+5 1 The denominator equals zero when x = -5. There is a vertical asymptote at x = -5. The domain is (-, -5)U(-5, )
4 Find the vertical asymptotes and domain of: g(x) = x 2 x 2 1 g(x) = x 2 (x+1)(x 1) Factor the denominator The denominator equals zero when x = -1 and x = 1. There are vertical asymptotes at x = -1 and x = 1. The domain is: (-, -1)U(-1, 1)U(1, )
5 Find the vertical asymptotes and domain of: h(x) = x2 +3x+2 x 2 +2x h(x) = (x+1)(x+2) x(x+2) h(x) = (x+1) x Factor the numerator and denominator Cancel common factors The denominator equals zero when x = 0. There is a vertical asymptote at x = 0. The domain is: (-, -2)U(-2, 0)U(0, )
6 Holes in a graph What happened at x = -2 in the previous example? The function is undefined at x = -2 because (x+2) was one of the factors of the denominator. Rational functions will have a hole in their graph at x-values where factors cancel out. h(x) = (x+1)(x+2) x(x+2) Hole at x = -2
7 Horizontal Asymptotes If a rational function is written in standard form: f(x) = a x h + k it will have a horizontal asymptote at y = k. If a rational function is written as f(x) = Polynomial, then Polynomial 1. If the numerator has a higher degree, it does not have a horizontal asymptote. 2. If the denominator has a higher degree, there is a horizontal asymptote of y = If the numerator and denominator have the same degree, there is a horizontal asymptote at: y = leading coefficient of numerator leading coefficient of denominator
8 The range of a rational function is all y values except horizontal asymptotes and holes. Ex) Determine the horizontal asymptote and range for: f(x) = 1 x 3 2 The rational function is written in standard form. There is a horizontal asymptote at y = -2. The range of the function is (-, -2)U(-2, )
9 Determine the horizontal asymptote and range for: g(x) = 3x2 4x+1 9x The degree of the numerator and denominator are the same. h(x) = 5x5 32x 2 12x 4 3x The degree of the numerator is greater than the denominator. There is a horizontal asymptote at y = 3 9 = 1 3. The range of the function is: (-, 1 3 )U(1 3, ) The function has no horizontal asymptote. The is a hole at (0, 0) The range of the function is: (-, 0) U(0, )
10 Determine the horizontal asymptote and range for: g(x) = x2 +2x+2 2x 3 +5 The degree of the denominator is greater than the numerator. There is a horizontal asymptote at y = 0. The range of the function is: (-, 0)U(0, ) h(x) = 6x 4 5x 4 3x 3 +2x 1 The degree of the numerator and denominator are the same. The function has a horizontal asymptote at y = 6 5 The range of the function is: (-, 6 5 )U( 6 5, ).
11 Continuity A function is continuous if there are no gaps, breaks, or holes in the graph. A function is discontinuous if there is at least one gap, break, or hole in the graph. Discontinuous Continuous
12 Continuous Discontinuous
13 Zeros The zeros (where the graph crosses the x-axis) are found where the numerator equals zero. You must simplify the rational function first. Ex) Find the zeros of f(x) = x2 7x+6 x 2 36 f(x) = (x 6)(x 1) (x+6)(x 6) f(x) = (x 1) (x+6) Factor the numerator and denominator Cancel common factors f(x) has a zero at x = 1.
14 Identify the asymptotes, holes, zeros, domain, range, and graph: g(x) = 3x2 +13x+4 x 2 +5x+4 g(x) = (3x+1)(x+4) (x+4)(x+1) g(x) = 3x+1 x+1 Factor the numerator and denominator Cancel common factors The function has a horizontal asymptote at y = 3 1 = 3, a vertical asymptote at x = -1, a hole at (-4,3.66), the domain is (-, -4)U(-4, -1)U(-1, ) the range is (-, 3)U(3, 3.66)U(3.66, )
15 x Identify the asymptotes, holes, zeros, domain, range, and graph: g(x) = 3x+1 x+1 g(x) /
16 HW pg 597 # s 20-31, 33-38
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