33. Conic Sections in General Form

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Presentation transcript:

33. Conic Sections in General Form H E 33. Conic Sections in General Form P

General Form of a Conic Equation We sometimes see conic equations written in General Form: where A, B, C, D, E and F are integers and A, B and C are NOT ALL equal to zero.

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)

What values form a circle? What values form an Ellipse? The values of the coefficients in the conic equation determine the TYPE of conic. What values form a circle? What values form an Ellipse? What values form a Hyperbola? What values form a Parabola?

Circles... where A & C have the SAME SIGN and they have the SAME VALUE NOTE: There is no Bxy term, and D, E & F may equal zero! For example:

Ellipses... where A & C have the SAME SIGN but DIFFERENT VALUES NOTE: There is no Bxy term, and D, E & F may equal zero! For example:

Hyperbola... where A & C have DIFFERENT signs. For example: NOTE: There is no Bxy term, and D, E & F may equal zero! For example:

Parabola... A Parabola can be oriented 2 different ways: 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:

Parabola …Vertical The following equations all represent vertical parabolas in general form; they all have a squared x term and a linear y term:

Parabola …Horizontal The following equations all represent horizontal parabolas in general form, they all have a squared y term and a linear x term:

Summary ... 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. 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