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2-5: Absolute Value Functions and Graphs

2-5: Absolute Value Functions and Graphs. Essential Question: What is the shape of the graph of a function of the form y = | mx + b|?. 2-5: Absolute Value Functions and Graphs. Absolute Value Functions: Are in the form f(x) = |mx + b| + c Are shaped like a “V”

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2-5: Absolute Value Functions and Graphs

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  1. 2-5: Absolute Value Functions and Graphs Essential Question: What is the shape of the graph of a function of the form y = |mx + b|?

  2. 2-5: Absolute Value Functions and Graphs • Absolute Value Functions: • Are in the form f(x) = |mx + b| + c • Are shaped like a “V” • Graphs will open up or down, depending on if it starts with a negative sign f(x) = |x| f(x) = -|x|

  3. 2-5: Absolute Value Functions and Graphs • Opening Up/Down • Determine whether the following absolute value functions open up (like a “V”) or open down (like a teepee) • y = |2x – 4| • y = -|x – 5| + 3 • y = -3|x + 4| - 6 • y = ½|-4x + 1| Opens Up Opens Down Opens Down Opens Up

  4. 2-5: Absolute Value Functions and Graphs • All absolute value graphs have a vertex • The vertex is at the point of the “V” • The vertex of y = |mx + b| + c is located at • Shortcut: • The x coordinate is found by setting the portion inside the absolute value = 0 and solving for x. • The y coordinate is the number that is outside the absolute value.

  5. 2-5: Absolute Value Functions and Graphs • Finding the vertex • Find the vertex of the function y = -|x + 1| - 2 • x-coordinate • x + 1 = 0 - 1 - 1 • x = -1 • y-coordinate • y = -2 • Vertex is at (-1, -2)

  6. 2-5: Absolute Value Functions and Graphs • Finding the vertex • Determine the vertex of the following absolute value graphs • y = |2x – 4| • y = -|x – 5| + 3 • y = -3|x + 4| - 6 • y = ½|-4x + 1| (2, 0) (5, 3) (-4, -6) (1/4, 0)

  7. 2-5: Absolute Value Functions and Graphs • Graphing an Absolute Value Function • Method #1: • Find the vertex • Choose two x-values (one less, one greater) than the x-value of the vertex. • Plug those two x-values into the absolute value function, and make two more points • Plot all three points, connected at the vertex.

  8. 2-5: Absolute Value Functions and Graphs • Graphing an Absolute Value Function • Example: • Graph y = -|x + 1| - 2 • Vertex is at (-1, -2) • Choose x-values of 0 and -2, plug into the equation • -|(0)+1|- 2 -|(-2)+1| - 2-|1| - 2 -|-1| - 2-1 – 2 -1 – 2-3 -3Point: (0, -3) Point: (-2, -3)

  9. 2-5: Absolute Value Functions and Graphs • Graphing an Absolute Value Function • Example: • Graph y = -|x + 1| - 2 • Vertex is at (-1, -2) • Point: (0, -3) Point: (-2, -3)

  10. 2-5: Absolute Value Functions and Graphs • Graphing an Absolute Value Function • Method #2: • Use the vertex and the slope. • The slope of the line is found by multiplying anything outside the absolute value by the number in front of the x. • Regardless of whether the slope is positive or negative, use the “opens up/opens down” to determine which way to go.

  11. 2-5: Absolute Value Functions and Graphs • Graphing an Absolute Value Function • Example: • Graph y = ½|-½ x + 2| - 2 • -½ x + 2 = 0 • -½ x = -2 • x = 4 y = -2 • Find the vertex: (4,-2) • Find the slope: y = ½|-½ x + 2| - 2 • The graph opens up • Slope is ¼ (ignore negative sign)

  12. 2-5: Absolute Value Functions and Graphs • Assignment • Page 90 • Problems 1-9 & 19 – 27 (odd problems) • Ignore instructions about making a table or sketching two lines… use whichever method we discussed today that you feel most comfortable with.

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