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Range from a graph:

Range is the set of y values that the function uses. List the bottom value, then the top value. It uses a parentheses ( ) if the value is an infinity symbol,or a square bracket [ ] if it is a number value that is included in the range.

 Piecewise function: The space used by the graphed function goes from -4 on the y axis to 3 on the y axis. The range is [-4,3].  Quadratic function: The space used by the graphed function heads to - ∞ on the bottom to 2 on the top. The range is (- ∞,2].
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Polynomial: You are looking for the lowest bottom and highest top when you write up the range. Check the left and right sides to see what infinities they head towards. Here, both head to positive infinity so that's the top. The lowest graphed part bottoms at -1. The range is [-1, ∞).  Exponential: The graph will have a long flat portion that approaches a y value without reaching it (called horizontal asymptote). The other end will head towards an infinity. Here, the horizontal asymptote is at -1 and it heads to postive infinity. The range is (-1, ∞).
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 Logarithms are the inverse functions of exponential functions.They have a vertical asymptote which limits domain. There is no limit on range. Write (- ∞,∞) for range. Graphed functions can be confusing because one side is so flat, but it keeps gradually creeping up or down forever.   Rationals break into pieces, so you write the range with restrictions. The restrictions are the y values it doesn't use. Here,it will use all y values except 0. Write it up as a horizontal asymptote. Horizontal asmyptote at y=0.
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Range from an equation:

Quadratics and Polynomials:

You'll need the y coordinate of the vertex or lowest turning point. Click here for how to do that.
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You need the horizontal asymptote to write up range. The value is whatever constant value is added or subtracted to the function outside of the exponential term. Look to see if it's a positive or negative exponential term. If it's positive, it's the value at the bottom, positive infinity. If the exponential term has a negative coefficient (out front), it approaches the horizontal asymptote from negative infinity. Write negative infinity, then the value.

Rational functions:

Horizontal asymptotes affect end behavior. To identify the y value, compare the exponents of the largest term when it's written in standard form. Standard form is when all the factor terms are multiplied together. Three rules to figure it out, depending on whether the top has a larger exponent, the bottom has a larger exponent, or the exponents are the same.
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rational exponents same rational exponent bigger bottom
Largest exponent same:The largest term in the numerator and denominator is the x, which has an exponent of 1. The value for the horizontal asymptote will be the ratio of coefficients of the two largest terms.Here, y=3/2. Largest exponent bottom: The largest exponent in the numerator is a 1, and the largest exponent in the denominator is a 2. The bottom has a bigger exponent. If the bottom is bigger, the value is a zero. y=0.
rational exponent top bigger
Largest exponent top:The numerator's largest exponent is a 3. The denominator just has an x,so it's exponent is a 1. When the numerator has a larger exponent, there is no horizontal asymptote. The function can use all values on the y axis.