Warm-Up 1. On approximately what interval is the function is decreasing. Are there any zeros? If so where? Write the equation of the line through.

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Presentation transcript:

Warm-Up 1. On approximately what interval is the function is decreasing. Are there any zeros? If so where? Write the equation of the line through the points (-3, 17) and (-15, 29).

Lesson 2.1: Graphing Absolute Value Functions We will be working on the following: Translating f(x) = |x| up, down, left and right Stretching and compressing f(x) = |x| vertically and horizontally Reflecting f(x) = |x| over the x-axis and y-axis Combinations of the above

Vertical and Horizontal Translations Given the parent function f(x) = |x|, f(x) = |x| + k is a vertical translation. If k is positive, translate up k units. If k is negative, translate down k units. f(x) = |x - h| is a horizontal translation. If h is positive, translate right h units. If h is negative, translate left h units.

Example For each of the following, describe how the parent function f(x) = |x| is transformed. f(x) = |x| - 6 This is a translation down 6 units. f(x) = |x+5| This is a translation left 5 units.

Vertical Stretch or Compression Given the parent function f(x) = |x|, f(x) = a|x| is a vertical stretch or compression by a factor of a. If a > 1, then vertical stretch. If 0 < a < 1, then vertical compression

Horizontal Stretch or Compression Given the parent function f(x) = |x|, f(x) = |bx| is a horizontal stretch or compression by a factor of . If b > 1, then horizontal compression. If 0 < b < 1, then horizontal stretch.

Example For each of the following, describe how the parent function f(x) = |x| is transformed. f(x) = 5|x| This is a vertical stretch by a factor of 5. f(x) = | x| This is a horizontal stretch by a factor of 2.

Reflection over x-axis or y-axis Given the parent function f(x) = |x|, f(x) = -|x| is a reflection over x-axis. f(x) = |-x| is a reflection over y-axis.

Example For each of the following, describe how the parent function f(x) = |x| is transformed. f(x) = -|x| + 3 This is a reflection over x-axis and then translated up 3 units. f(x) = 5|x+3| - 7 This is a vertical stretch by a factor of 5 and then a translation 3 units to the left and 7 units down.