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Warm-up Complete the 5 question quiz for your warm-up today

MA3A1. Students will explore rational functions. a. Investigate and explain characteristics of rational functions, including domain, range, zeros, points of discontinuity, intervals of increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior. b. Find inverses of rational functions, discussing domain and range, symmetry, and c. Solve rational equations and inequalities analytically, graphically, and by using appropriate technology.

Graphs & Characteristics of Rational Functions By the end of today, you will know how to find the following characteristics algebraically: 1.Domain and Range 2.x-intercepts and y-intercepts 3.Horizontal and vertical asymptotes 4.Holes ** Important Note: Rational Functions should always be FACTORED before you do ANYTHING! If you skip this step, then you will probably do more work than you needed to!

You WILL be able to take the following problem and make a list of these characteristics: Holes: Domain: Range: x-intercepts: y-intercepts: Horizontal asymptotes: Vertical asymptotes:

Hole If you can cancel a factor in both the numerator and the denominator, then a rational function has a hole. To find the x-coordinate, set the canceled factor equal to zero and solve. Then plug that value into “the leftovers” to find the y-coordinate. Example: HOLE: (3, 1/7)

Determine if the function has a hole. If so, find the coordinates.

Domain is all real numbers except the vertical Asymptote and the x-value of the hole.

Find the Domain.

A rational function has a vertical asymptote at each value of x that makes only the denominator equal zero. It’s value is the numbers you used to state the domain restrictions! written as x = # How to find the Vertical Asymptotes: Set your denominator equal to zero and solve. You will need to exclude the x-value of the hole if applicable. Vertical Asymptotes:Domain:

Find the Vertical Asymptotes:

Find the Horizontal Asymptotes:

All real numbers except the horizontal asymptote

Find the Range:

To find the x-int of Rational Functions, set the numerator equal to zero and solve for x. Or look at the graph and see where the graph touches/intersect the x-axis.

Find all x-intercepts of each function.

To find the y-int of Rational Functions, substitute 0 for x. Or look at the graph and see where the graph touches the y-axis.

Find all y-intercepts of each function.