Key Terms 3.1 polynomial function leading coefficient dominating term zero polynomial quadratic function parabola axis of symmetry (axis) vertex quadratic regression 3.2 synthetic division zero of a polynomial function root (or solution) of an equation 3.3 multiplicity of a zero 3.4 turning points end behavior 3.5 rational function discontinuous graph vertical asymptote horizontal asymptote oblique asymptote point of discontinuity (hole) 3.6 nonlinear inequality polynomial inequality 3.7 varies directly (directly proportional to) constant of variation varies inversely (inversely proportional to) combined variation varies jointly New Symbols z conjugate of z = a + bi , , , end behavior diagrams ∣ ƒ1x2 ∣ uH absolute value of ƒ1x2 increases without bound xua x approaches a Chapter 3Test Prep 420 CHAPTER 3 Polynomial and Rational Functions Quick Review Concepts Examples Graph ƒ1x2 = -1x + 322 + 1. The graph opens down because a 60. It is the graph of y = -x2 shifted to the left 3 units and up 1 unit, so the vertex is 1-3, 12, with axis x = -3. The domain is 1-∞, ∞2, and the range is 1-∞, 14. The function is increasing on 1-∞, -32 and decreasing on 1-3, ∞2. Graph ƒ1x2 = x2 + 4x + 3. The vertex of the graph is a- b 2a , ƒ a- b 2abb = 1-2, -12. a = 1, b = 4, c = 3 The graph opens up because a 70. ƒ102 = 3, so the y-intercept is 10, 32. The solutions of x2 + 4x +3 = 0 are -1 and -3, which correspond to the x-intercepts. The domain is 1-∞, ∞2, and the range is 3-1, ∞2. The function is decreasing on 1-∞, -22 and increasing on 1-2, ∞2. x y 0 3 –3 1 (–2, –1) x = –2 –1 f(x) = x2 + 4x + 3 3.1 Quadratic Functions and Models 1. The graph of ƒ1x2 =a1x −h22 +k, with a≠0, is a parabola with vertex 1h, k2 and axis of symmetry x = h. 2. The graph opens up if a 70 and down if a 60. 3. The graph is wider than the graph of ƒ1x2 = x2 if a 61 and narrower if a 71. Vertex Formula To find the vertex of the graph of ƒ1x2 = ax2 + bx + c, with a≠0, complete the square or use the vertex formula. a− b 2a , ƒ a− b 2abb Vertex Graphing a Quadratic Function ƒ1x2 =ax2 +bx +c Step 1 Find the vertex using the vertex formula or by completing the square. Plot the vertex. Step 2 Plot the y-intercept by evaluating ƒ102. Step 3 Plot any x-intercepts by solving ƒ1x2 = 0. Step 4 Plot any additional points as needed, using symmetry about the axis. The graph opens up if a 70 and down if a 60. x y 0 1 –2 –5 1 (–3, 1) x = –3 f(x) = –(x + 3)2 + 1
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