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## Identifying Conic Sections

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**Identifying Conic Sections**How do I determine whether the graph of an equation represents a conic, and if so, which conic does it represent, a circle, an ellipse, a parabola or a hyperbola? Created by K. Chiodo, HCPS**We usually see conic equations written in General, or**Implicit Form: General Form of a Conic Equation where A, B, C, D, E and F are integers and A, B and C are NOT ALL equal to zero. Note: You may see some conic equations solved for y, but if the equation can be re-written into the form above, it is a conic equation!**Please Note:**A conic equation written in General Form doesn’t have to have all SIX terms! Several of the coefficients A, B, C, D, E and F can equal zero, as long as A, B and C don’t ALL equal zero. If A, B and C all equal zero, what kind of equation do you have? ... T H I N K... Linear!**So, it’s a conic equation if...**• the highest degree (power) of x and/or y is 2 (at least ONE has to be squared) • the other terms are either linear, constant, or the product of x and y • there are no variable terms with rational exponents (i.e. no radical expressions) or terms with negative exponents (i.e. no rational expressions)**The values of the coefficients in the conic equation**determine the TYPE of conic. What values form an Ellipse? What values form a Hyperbola? What values form a Parabola?**Ellipses...**where A & C have the SAME SIGN NOTE: There is noBxy term, and D, E & Fmay equal zero! For example:**This is an ellipse since x & y are both squared, and both**quadratic terms have the same sign! Center (-2, 0) Hor. Axis = 2 Vert. Axis = √8 The General, or Implicit, Form of the equations can be converted to Graphing Form by completing the square and dividing so that the constant = 1. Ellipses...**Vert. axis = 2/√3**center (-1, 1) Hor. axis = 2 In this example, x2 and y2 are both negative (still the same sign), you can see in the final step that when we divide by negative 4 all of the terms are positive. Ellipses...**Radius = √5**Center (-2, 0) Ellipses…a special case! When A & C are the samevalue as well as the same sign, the ellipse is the same length in all directions … Circle! it is a ...**Hyperbola...**where A & C have DIFFERENT signs. NOTE: There is noBxy term, and D, E & Fmay equal zero! For example:**This is a hyperbola since x & y are both squared, and the**quadratic terms have different signs! y-axis=3 x-axis=2 Center (2,-1) The General, or Implicit, Form of the equations can be converted to Graphing Form by completing the square and dividing so that the constant = 1. Hyperbola...**x-axis=2**Center (0,3) y-axis=2 In this example, the signs change, but since the quadratic terms still have different signs, it is still a hyperbola! Hyperbola...**A parabola is vertical if the equation has an x squared term**AND a linear y term; it may or may not have a linear x term & constant: A parabola is horizontal if the equation has a y squared term AND a linear x term; it may or may not have a linear y term & constant: A Parabola can be oriented 2 different ways: Parabola...**Parabola …Vertical**The following equations all represent vertical parabolas in general form; they all have a squared x term and a linear y term:**Vertex (2,3)**Parabola …Vertical To write the equations in Graphing Form, complete the square for the x-terms. There are 2 popular conventions for writing parabolas in Graphing Form, both are given below:**Vertex (-1,4)**Parabola …Vertical In this example, the signs must be changed at the end so that the y-term is positive, notice that the negative coefficient of the x squared term makes the parabola open downward.**Parabola …Horizontal**The following equations all represent horizontal parabolas in general form, they all have a squared y term and a linear x term:**Vertex (1,-4)**Parabola …Horizontal To write the equations in Graphing Form, complete the square for the y-terms. There are 2 popular conventions for writing parabolas in Graphing Form, both are given below:**Vertex (3,0)**Parabola …Horizontal In this example, the signs must be changed at the end so that the x-term is positive; notice that the negative coefficient of the y squared term makes the parabola open to the left.**What About the term Bxy?**None of the conic equations we have looked at so far included the term Bxy. This term leads to a hyperbolic graph: or, solved for y:**where A, B, C, D, E and F are integers and A, B and C are**NOT ALL equal to zero. Summary ... General Form of a Conic Equation: Identifying a Conic Equation:**Practice ...**Identify each of the following equations as a(n): (a) ellipse (b) circle (c) hyperbola (d) parabola (e) not a conic See if you can rewrite each equation into its Graphing Form!**Answers ...**(a) ellipse (b) circle (c) hyperbola (d) parabola (e) not a conic**P**H Conic Sections ! C E Created by K. Chiodo, HCPS