![]() ![]() Students will be able to Identify and Graph piecewise functions (including absolute value) as evidenced by a math journal. □ means −(□)AND □ since they are both the same distance from zero. As we progress through this course we will see what || means in higher dimensions. In one dimension we usually say “|−3| means how far is −3 away from zero?” and we say 3, so |−3|=3. What is absolute value? Absolute value || is the SIZE or a number (one, two, three dimensional). Know the difference between the symbols means NOT included so we use and OPEN circle ≤≥ means included, so we use a CLOSED circle Understand what the boundaries mean, four types □>□ all values to the right of a □3 How do we graph piecewise functions? Use multiple colors if you have them, one color for each piece. Need at least two points Linear □□+□, what is the slope? What is the y intercept? Need at least 2 points Quadratic □ □ 2 +□□+□ or □ □−ℎ 2 +□ whose vertex is (ℎ, □) and □ determines directions of opening (up or down) and compression (wide or skinny) Need at least three points Cubic □ □ 3 where □ is the directions of opening and compression, need three points What common functions are used to “piece” together? Constant Just a number, looks like a horizontal line. “Frankenstein Functions”ģ What common functions are used to “piece” together? What is a piecewise function? A piecewise function (passes vertical line test) is a function that is made up of different graphs for different intervals. After plotting a number of points the general shape of the reciprocal function can be determined.1 1.7 Piecewise Functions Students will be able to Identify and Graph piecewise functions (including absolute value) as evidenced by a math journal. This describes a horizontal asymptote A horizontal line to which a graph becomes infinitely close where the x-values tend toward ±∞. In other words, as the x-values become very large the resulting y-values tend toward zero. Furthermore, where the x-values are very large the result of the reciprocal function is very small.į ( 10 ) = 1 10 = 0.1 f ( 100 ) = 1 100 = 0.01 f ( 1000 ) = 1 1,000 = 0.001 ![]() This describes a vertical asymptote A vertical line to which a graph becomes infinitely close. ![]() In other words, as the x-values approach zero their reciprocals will tend toward either positive or negative infinity. The reciprocal of an x-value very close to zero is very large. , defined by f ( x ) = 1 x, is a rational function with one restriction on the domain, namely x ≠ 0. The reciprocal function The function defined by f ( x ) = 1 x. The domain and range both consist of real numbers greater than or equal to zero [ 0, ∞ ).
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