Graphing Rational Functions RATEY Graphing Rational Functions Using

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Graphing Rational Functions “RATEY”

Graphing Rational Functions “RATEY”

Graphing Rational Functions Using “RATEY” R – Roots A – Asymptotes T – Tangent/Together

Graphing Rational Functions Using “RATEY” R – Roots A – Asymptotes T – Tangent/Together E – End Behavior Y – Y-Intercept

ROOTS – Solution to TOP These are the x-intercepts of the graph Set the

ROOTS – Solution to TOP These are the x-intercepts of the graph Set the numerator (top of fraction) = 0 and solve for x Write the answers as points: (# , 0) Graphs MUST pass through or touch any roots on the graph and CANNOT cross the x-axis anywhere else

ASYMPTOTES (Vertical) – Solution to Bottom These are the Vertical Asymptotes Set the denominator

ASYMPTOTES (Vertical) – Solution to Bottom These are the Vertical Asymptotes Set the denominator (bottom of the fraction) = 0 and solve for x Write the answers as lines x = # A vertical asymptote acts as a boundary line The graph can come close to, but never cross, this line!!

These give clues about special behaviors of the graph Tangent – ON TOP: Factors

These give clues about special behaviors of the graph Tangent – ON TOP: Factors with multiplicities of 2 means the graph is tangent to the x-axis (touches the x-axis). Together – ON BOTTOM: Factors with multiplicities of 2 means the graph will come together at that asymptote.

Factors with Multiplicities of 2 : An x² term alone (monomial) or a factor

Factors with Multiplicities of 2 : An x² term alone (monomial) or a factor squared, like (x + 5)² Examples: YES: x², -2 x², (x – 5)², (x + 3)² NO: x² + 2 x, - x² + 3, 2 x² + x – 5

END BEHAVIOR – Look at RATIO of FIRST TERMS This tells how the graph

END BEHAVIOR – Look at RATIO of FIRST TERMS This tells how the graph looks on the ends – to the left and right Write out the first terms as a RATIO. Use the degree of the top and bottom to decide on one of the three categories below, and then reduce the first terms. This is the same idea as when we looked at end behavior for Polynomial Functions by examining the Leading Term of the Polynomial. For Rational Functions we have two Polynomials (one on TOP and one on BOTTOM). So we use the Power Functions in a RATIO, and simplify.

Y-INTERCEPT – Plug in 0 for x This is the y-intercept of the graph

Y-INTERCEPT – Plug in 0 for x This is the y-intercept of the graph Plug in 0 for x and simplify. Write this as the point (0, #)

RATEY Examples - #1

RATEY Examples - #1

RATEY Examples - #1 Set top factors equal to 0 and solve for x

RATEY Examples - #1 Set top factors equal to 0 and solve for x x=0 There is a root at (0, 0) Set bottom factors equal to 0 and solve for x x+3=0 x = -3 There is a vertical asymptote at x = -3 Squared Factors on Top – Tangent Squared Factors on Bottom – Together This equation has neither Check degrees – This equation is “balanced” so we divide the first terms. Y = x/x = 1 The end behavior follows the vertical asymptote y=1 Make all x values 0 and simplify. Y = 0/(0+3) = 0 Y-intercept at (0, 0)

RATEY Examples - #2

RATEY Examples - #2

RATEY Examples - #2 Set top factors equal to 0 and solve for x

RATEY Examples - #2 Set top factors equal to 0 and solve for x 5=0 this is a false statement There are no roots Set bottom factors equal to 0 and solve for x X-2=0 x = 2 There is a vertical asymptote at x = 2 Squared Factors on Top – Tangent Squared Factors on Bottom – Together This is together at the asymptote x=2 Check degrees – This equation is “bottom heavy” so the end behavior follows a horizontal asymptote of y=0 Make all x values 0 and simplify. Y = 5/(0 -2)2 = 5/4 = 1. 25 Y-intercept at (0, 1. 25)

ASSIGNMENT RATEY Worksheet #1

ASSIGNMENT RATEY Worksheet #1