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12.1 Parabolas 12.2 Circles 12.3 Ellipses 12.4 Hyperbolas

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12.1 Parabolas 12.2 Circles 12.3 Ellipses 12.4 Hyperbolas 8422d_c12_946-970 6/6/02 6:17 PM Page 946 mac85 Mac 85:365_smm:8422d_Thomasson: 12.1 Parabolas 12.2 Circles 12.3 Ellipses 12.4 Hyperbolas 8422d_c12_946-970 6/6/02 6:18 PM Page 947 mac85 Mac 85:365_smm:8422d_Thomasson: Chapter mooth, regular curves are common in the world around us. The orbit of the space shuttle around the SEarth is one kind of smooth curve; another is the shape of domed buildings such as the United States Capitol. Conic Television receiving dishes are a very common curved shape, as are the curved rearview mirrors on your car. Sections Mathematically, however, an interesting property of all the curves just mentioned is that they are expressed as qua- dratic equations in two variables. We discussed quadratic equations in Chapter 5. In the current chapter, we extend that discussion to other forms of equations and their related graphs, called conic sections: parabolas, circles, ellipses, and hyperbolas. We will describe each curve and analyze the equation used to graph it. We will also discuss properties that make these curves so use- ful in so many different areas, from engineering to archi- tecture to astronomy. We conclude the chapter with a project that further explores the various conic sections we have studied. 12.1 APPLICATION A satellite dish receiver is in the shape of a parabola. A cross section of the dish shows a diameter of 13 feet at a distance of 2.5 feet from the vertex of the parabola. Write an equation for the parabola. After completing this section, we will discuss this application fur- ther. See page 000. 8422d_c12_946-970 6/6/02 6:18 PM Page 948 mac85 Mac 85:365_smm:8422d_Thomasson: 948 Chapter 12 • Conic Sections 12.1 Parabolas OBJECTIVES 1 Write quadratic equations in the form y ϭ a x Ϫ h 2 ϩ k. 1 2 2 Understand the effects of the constants h and k on the graph of a quadratic equation in the form y ϭ x Ϫ h 2 ϩ k. 1 2 3 Graph vertical parabolas. 4 Graph horizontal parabolas. 5 Write a quadratic equation for a parabola, given its vertex and a point on the curve. 6 Model real-world situations by using parabolas. Conic sections are the curves obtained by intersecting a plane and a right cir- cular cone. A plane perpendicular to the cone’s axis cuts out a circle; a plane parallel to a side of the cone produces a parabola; a plane at an arbitrary angle to the axis of the cone forms an ellipse; and a plane parallel to the axis cuts out a hyperbola. If we extend the cone through its vertex and form a second cone, you find the second branch of the hyperbola. All these curves can be described as graphs of second-degree equations in two variables. CircleParabola Ellipse Hyperbola We have described the shape of a graph of a quadratic equation as a para- focus bola. In this section, we define that curve geometrically. A parabola is defined P as the set of all points in a plane that are equidistant from a line and a point not on the line. The line is called the directrix and the point is called the focus directrix (plural, foci). 12.1.1 Writing Quadratic Equations in the Form y ϭ a x Ϫ h 2 ϩ k 1 2 The standard form of a quadratic function is y ϭ ax2 ϩ bx ϩ c. In this section, we will write the quadratic equation in the form y ϭ a x Ϫ h 2 ϩ k, where 1 2 a, h, and k are real numbers. To do that, we will need to complete the square. 8422d_c12_946-970 6/6/02 6:18 PM Page 949 mac85 Mac 85:365_smm:8422d_Thomasson: 12.1 • Parabolas 949 To write a quadratic equation in the form y ϭ a x Ϫ h 2 ϩ k, 1 2 • Isolate the x-terms to one side of the equation. • Factor out the leading coefficient. • Add the value needed to complete the square to both sides of the equation. • Rewrite the trinomial as a binomial squared. • Solve the equation for y. EXAMPLE 1 Write the quadratic equations in the form y ϭ a x Ϫ h 2 ϩ k. Identify a, h, 1 2 and k. a.y ϭ x2 Ϫ 4x ϩ 7 b. y ϭϪ2x2 Ϫ 16x Ϫ 35 Solution a. y ϭ x2 Ϫ 4x ϩ 7 2 y Ϫ 7 ϭ x Ϫ 4x Isolate the x-terms. 2 Ϫ y Ϫ 7 ϩ 4 ϭ x Ϫ 4x ϩ 4 Add 4 2 ϭ 4 to both sides. 1 2 2 2 y Ϫ 3 ϭ x Ϫ 2 Write the trinomial as a binomial squared. 1 2 2 y ϭ x Ϫ 2 ϩ 3 Solve for y. 1 2 In the equation y ϭ x Ϫ 2 2 ϩ 3, a ϭ 1, h ϭ 2, and k ϭ 3. 1 2 b. y ϭϪ2x2 Ϫ 16x Ϫ 35 2 y ϩ 35 ϭϪ2x Ϫ 16x Isolate the x-terms. 2 y ϩ 35 ϭϪ2 x ϩ 8x Factor out the leading coefficient, 1 2 Ϫ2. 2 y ϩ 35 ϩ Ϫ2 16 ϭϪ2 x ϩ 8x ϩ 16 Add Ϫ2 8 2, or Ϫ2 16 , to both 1 2 2 1 2 3 1 24 1 2 sides. 2 y ϩ 3 ϭϪ2 x ϩ 4 Write the trinomial as a binomial 1 2 squared. 2 y ϭϪ2 x ϩ 4 Ϫ 3 Solve for y. 1 2 In the equation y ϭϪ2 x ϩ 4 2 Ϫ 3, or y ϭϪ2 x Ϫ Ϫ4 2 ϩ Ϫ3 , 1 2 3 1 24 1 2 a ϭϪ2, h ϭϪ4, and k ϭϪ3. • HELPING HAND Note that the leading coefficient a is also the same value as ( the real number a in our new equation. 12.1.1 Checkup In exercises 1 and 2, write the quadratic equations in the form 3. When you rewrite a quadratic equation in the form y ϭ a x Ϫ h 2 ϩ k. Identify a, h, and k. y ϭ a x Ϫ h 2 ϩ k, is the new equation equivalent to the 1 2 1 2 1.y ϭ x2 ϩ 6x ϩ 7 2. y ϭ 3x2 Ϫ 12x ϩ 5 original equation? Explain. 12.1.2 Understanding the Effects of the Constants In Chapter 5, we described the graph of a quadratic equation in two variables, y ϭ f x ϭ ax2 ϩ bx ϩ c, as a parabola. We discovered that the values of the 1 2 real numbers a, b, and c affected the shape of the graph. 8422d_c12_946-970 6/6/02 6:18 PM Page 950 mac85 Mac 85:365_smm:8422d_Thomasson: 950 Chapter 12 • Conic Sections If the quadratic equation is written in the form y ϭ a x Ϫ h 2 ϩ k, do the 1 2 values of the real numbers a, h, and k affect the shape of the parabola? To find out, complete the following set of exercises. Discovery 1 Effect of the Real Numbers h and k of a Quadratic Equation on a Parabola 1. Sketch the graphs of the following quadratic equations of the form y ϭ a x Ϫ h 2 ϩ k, where a ϭ 1 and h ϭ 0. Label the vertex of each 1 2 graph. y ϭ x2 ϩ 2 y ϭ x2 Ϫ 2 2. Write a rule for determining the y-coordinate of the vertex of a parabola from the equation of the parabola. 3. Sketch the graphs of the following quadratic equations of the form y ϭ a x Ϫ h 2 ϩ k, where a ϭ 1 and k ϭ 0. Label the vertex of each 1 2 graph. y ϭ x ϩ 3 2 y ϭ x Ϫ 3 2 1 2 1 2 4. Write a rule for determining the x-coordinate of the vertex of a parabola from the equation of the parabola. 5. Sketch the graph of the following quadratic equations of the form y ϭ a x Ϫ h 2 ϩ k, where a ϭ 1. Label the vertex of each graph. 1 2 y ϭ x ϩ 3 2 ϩ 2 y ϭ x Ϫ 3 2 Ϫ 2 1 2 1 2 6. Write a rule for determining the coordinates of the vertex of a parabola from the equation of the parabola. The vertex of the parabola is h, k . Since the x-coordinate of the vertex is h, 1 2 the axis of symmetry is the graph of the vertical line x ϭ h. The graph of a qua- dratic equation of the form y ϭ a x Ϫ h 2 ϩ k is called a vertical parabola, 1 2 because its axis of symmetry is a vertical line. Remember that the real number a is also the leading coefficient of the equa- tion y ϭ ax2 ϩ bx ϩ c. Therefore, it also affects the shape of the graph. SUMMARY OF THE EFFECTS OF THE REAL NUMBERS a, h, AND k OF A QUADRATIC EQUATION ON A VERTICAL PARABOLA The real numbers a, h, and k of a quadratic equation in the form y ϭ a x Ϫ h 2 ϩ k affect the graph of the equation. 1 2 If a 7 0, then the graph is concave upward (opens upward). If a 6 0, then the graph is concave downward (opens downward).
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