Introduction to Rational Equations
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Transcript of Introduction to Rational Equations
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Introduction to Rational Equations
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2 Types of Functions•Continuous•Discontinuous
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Continuous
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Continuous
• Keeps going• No breaks in graph• Smooth
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Discontinuous
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Discontinuous
• Stops• Graph has breaks or holes
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Examples
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• Continuous Graphs → Polynomials• Discontinuous Graphs → Rational
Equations
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Examples
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Your Turn: Be Prepared to Share!!!• Complete problems 1 – 6 on the Introduction
to Rational Equations handout• Remember, you need to:– Classify the graph as either continuous or
discontinuous– Classify the graph as either a polynomial or a
rational equation– Justify your reasoning!!!
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Sharing Activity1. I will gently throw the ball to a student.2. That student answers the first question.3. Then the student will gently throw the ball to
another student.4. That student answers the next question.5. Repeat until we’ve answered all the
questions.Say the student’s name before you throw
him/her the ball!
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Polynomial
• Monomial• Binomial• Trinomial
•Polygamy•Polytheism•Polydactyl•Polyglot
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Polynomials, cont.• A polynomial is an algebraic expression that
can be written in the formanxn + an-1xn-1 + … + a2x2 + a1x1 + a0
• An equation or an expression with a single variable raised to (usually many) powers
• All exponents are whole numbers• an ≠ 0 (Leading Coefficient ≠ 0)
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Polynomial Examples• Generally a long list of variables• f(x) = x4 – 4x3 + 2x2 – 3x + 11• f(x) = x11 + 7x5 – 4x3 + x – 12
• But we can also have a short list of variables• f(x) = x5 + x• f(x) = x2 – 1
• Or even no variables at all!• f(x) = 10• f(x) = ½
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Rational Equation
PolynomialPolynomialRational
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Rational Equations, cont.
• Rational equations are fractions in which both the numerator and the denominator are polynomials
• We don’t need variables in the numerator, but we must have them in the denominator!!!
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Rational Examples
423
2
xx
xx)x(f42
x
x)x(f
11
3
xx)x(f
4
2x)x(f
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Polynomials vs. Rational Equations
7. f(x) = x8 – 7x2 + 4 8. f(x) = 11
9. 10.
11.
162
3
xx)x(f x
2x
3x)x(f
23
x4
x1)x(f 2
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Your Turn: Be Prepared to Share!!!
• Complete problems 12 – 17 on the Introduction to Rational Equations handout.
• Remember, you need to:– Classify the equation as either a polynomial or a
rational equation.– Justify your reasoning
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Compare – Contrast – Summarize Graphic Organizer
Continuity → Continuous or Discontinuous
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How Alike?
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How Different?Polynomials
With Regard to Graphs
Rational Equations
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How Different?Polynomials
With Regard to Equations
Rational Equations
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How Different?Polynomials
With Regard to Continuity
Rational Equations
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Summarize:
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Discontinuous Graphs
Discontinuities
Rational Graphs
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*Discontinuities
• Discontinuity – a point or a line where the graph of an equation has a hole, a jump, a break, or a gap
• Affect the shape, domain and range of an equation
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Discontinuities, cont.
• Three major types of discontinuities:
• Vertical Asymptotes
• Horizontal Asymptotes
• Holes
Asymptotes
Point (Removable) Discontinuity
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Type of Discontinuities – Asymptotes• Lines that the graph
approaches but (almost) never crosses
• Represented by a dashed line
• Not part of the equation
• We don’t draw them if they happen on either the x-axis or the y-axis
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Vertical Asymptotes (1st Column)• Occur when the numerator is a non-zero # and
the denominator equals zero• Can never be crossed• Always in the form x = • Abbreviated VA
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Vertical Asymptotes, cont.Hand Drawn Calculator Drawn
The calculator doesn’t draw the asymptotes!!!!
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Experiment
• Graph in your graphing calculator1x
1y 2
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Calculators and Vertical Asymptotes
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Horizontal Asymptotes (2nd Column)• Occur when the degree of the denominator is ≥
the degree of the numerator
• Ex. • Can be crossed when |x| is very small• Describes the end behavior of a rational equation• Always in the form y = • Abbreviated HA
3xxy
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Horizontal Asymptotes, cont.Hand Drawn Calculator Drawn
The calculator doesn’t draw the asymptotes!!!!
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Point (Removable) Discontinuities – Holes (3rd Column)
• Gaps in the graph at a single point
– Occurs when
• Always in the form x =• Represented by an open circle (or hole) in
the graph
00
y
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Holes, cont.Hand Drawn
242
x
x)x(f
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Graphing Calculators and Holes• Graphing calculators have difficulty showing
removable discontinuities****Check the table for errors!
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Example #1
• x-int =
• y-int =
• VA:
• HA:
• Holes:
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Example #2
• x-int =
• y-int =
• VA:
• HA:
• Holes:
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Your Turn:
• Complete problems 1 – 6 on the Identifying Features of Rational Equations Practice handout.
• Don’t answer the domain and range questions!
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1. 2.
3. 4.
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5. 6.
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Discontinuities and Domain and Range
• Discontinuities affect the domain and range of a rational equation
• Vertical Asymptotes → Domain• Horizontal Asymptotes → Range• Holes → Domain and Range
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Example 1:
•
• Domain:
• Range:
3
2
xxy
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Example 2:
•
• Domain:
• Range:
242
x
xy
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Your Turn:
• Answer the domain and range questions for problems 1 – 6 on the Identifying Features of Rational Equations Practice handout.
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1. 2.
3. 4.
5. 6.
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Homework
• Complete problems 1 – 6 on the Identifying the Features of Rational Equations Homework handout.
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Exit Ticket• Identify the following
features of the graph on the right:– x-int. =– y-int. =– VA:– HA:– Holes:– Domain:– Range: