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Absolute Value Functions and Graphs Lesson 2-5

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Important Terms Parent function: the simplest function with these characteristics. The equations of the function in a family resemble each other, and so do the graphs. Offspring of parent functions include translations, stretches, and shrinks. Translation: it shifts a graph horizontally, vertically, or both. It results in a graph of the same shape and size but possibly in a different position Stretch: a vertical stretch multiplies all y-values by the same factor greater than 1, thereby stretching the graph vertically Shrink: a vertical shrink reduces y-values by a factor between 0 and 1, thereby compressing the graph vertically Reflection: in the x-axis changes y-values to their opposites. When you change the y-value of a graph to their opposites, the graph reflects across the x-axis (creates a mirror image)

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The Family of Absolute Value Functions Vertical Translation Parent functionY=|x|Y=f(x) Translation up k units, k>0Y=|x|+kY=f(x)+k Translation down k units, k<0Y=|x|-kY=f(x)-k Horizontal Translation Parent FunctionY=|x|Y=f(x) Translation right h units, h>0Y=|x-h|Y=f(x-h) Translation left h units, h<0Y=|x+h|Y=f(x+k) Combined Translation (right h units, up k units)Y=|x-h|+kY=f(x-h)+k

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Families of Functions: Absolute Value Functions Vertical Stretch or Shrink, and Reflection in x-axis Parent functionY=|x|Y=f(x) Reflection in x-axisY=-|x|Y= -f(x) Stretch (a>1)Y=a|x|Y=af(x) Shrink (0
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Absolute Value An Absolute Value graph is always in a “V” shape.

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Given the following function, If: a > 0, then shift the graph “a” units up If: a < 0, then shift the graph “a” units down

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Given the following function, Since a > 0, then shift the graph “3” units up

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Let’s Graph

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How will the graph look?

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Let’s Graph

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How will the graph look?

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Let’s Graph

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How will the graph look?

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Let’s Graph

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Given the following function, We get the expression (x - b) and equal it to zero x - b = 0 x = b If: b > 0, then shift the graph “b” units to the right If: b < 0, then shift the graph “b” units to the left

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Given the following function, x – 1 = 0 x = 1 Since 1 > 0, then shift the graph “1” unit right

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Let’s Graph

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How will the graph look?

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Let’s Graph

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How will the graph look?

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Let’s Graph

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How will the graph look?

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Let’s Graph

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Graphing Recall: Shift “3” units up since 3 > 0 then we use the expression x + 1, and equal it to zero x + 1 = 0 x = -1 Since –1 < 0, then we shift “1” unit to the left

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Let’s Graph

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How will the graph look?

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Let’s Graph

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How will the graph look?

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Let’s Graph

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How will the graph look?

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Let’s Graph

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Given the following function, For this equation, c determines how wide or thin it will be. if: |c|>1, then the graph is closer to the y-axis if: |c|=1, then the graph remains the same if: 0<|c|<1, then the graph is further from the y-axis if c is a negative number, then the graph will reflect on the x-axis

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Given the following function, Since |5| > 0, then the graph is closer to the y-axis

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Let’s Graph

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How will the graph look?

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Let’s Graph

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How will the graph look?

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Let’s Graph

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How will the graph look?

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Let’s Graph

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How will the graph look?

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Let’s Graph

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Given the following function, Since 4 > 0, shift the graph “4” units up x – 1 = 0 x = 1 Since 1 > 0, then shift the graph “1” unit to the right Since |5| > 0 shift the graph closer to the y-axis.

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Let’s Graph

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How will the graph look?

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Let’s Graph

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How will the graph look?

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Let’s Graph

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How will the graph look?

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Let’s Graph

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How will the graph look?

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Let’s Graph

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How will the graph look?

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Let’s Graph

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How will the graph look?

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Let’s Graph

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Let’s Graph

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Congratulations!! You just completed the transformation of

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