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Warm Up Find the solutions for each absolute value equation:

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**Graphing Absolute Value Equations**

Math 8H Graphing Absolute Value Equations Algebra Glencoe McGraw-Hill JoAnn Evans

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The ABSOLUTE VALUE of a real number is the distance between the origin and the point representing the real number. The number 5 is five spaces from 0, the origin. |x| = x when x > 0 0 is zero spaces from itself. |0| = 0 The number -3 is three spaces from 0, the origin. |x| = -x when x < 0 Distance is not negative; the absolute value of a number will never be negative.

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**Graph the equation: y = |x|**

x y Every absolute value equation will graph into a v-shape. The VERTEX is the point of the v-shaped graph. Some will open up, others will open down.

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**How does this graph differ from y = |x|?**

Graph y = -|x| Graph y = |x - 2| How does this graph differ from y = |x|? How does this graph differ from y = |x|? x y x y -2 -1 1 2 -2 -1 -2 2 4 6 4 2 vertex vertex

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**How does this graph differ from y = |x|? **

Graph y = |x| + 1 Graph y = |x| - 3 How does this graph differ from y = |x|? How does this graph differ from y = |x|? x y x y -2 -1 1 2 3 2 1 -2 -1 1 2 -1 -2 -3 vertex vertex

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**How does this graph differ from y = |x|? **

Graph y = |x + 2| Graph y = |x - 1| How does this graph differ from y = |x|? How does this graph differ from y = |x|? x y x y -4 -3 -2 -1 2 1 -1 1 2 3 2 1 vertex vertex

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**Why is this useful information**

It’s possible to tell what the x value of the vertex will be just by looking at the absolute value equation. The value of x that will make the expression INSIDE the absolute value symbol equal to ZERO will be the x-value of the vertex of the graph. Why is this useful information Knowing the x-value of the vertex will help you to efficiently select x-values for the table of values. You need several values on either side of the vertex in order to see the v-shape appear.

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**To Sketch the Graph of an Absolute Value Equation:**

Find the value of x that will make the expression inside the absolute value symbol equal to zero. Place this value of x in the middle of your table of values. Choose two values of x less than this number and two values of x greater than this number. Calculate the corresponding y values and sketch the resulting v-shaped graph. If the x values are evenly spaced on either side of the x value of the vertex, the y values should show a pattern.

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**-2 Sketch the graph of y = |x + 2| - 3**

What value of x will make the expression inside the absolute value sign equal to 0? -2 x y -4 -3 -2 -1 -1 -2 is the x value of the vertex. Place it in the middle of the table. Choose 2 values less and 2 values more, evenly spacing them. -2 -3 -2 -1

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**What value of x will make the expression inside the absolute value sign equal to 0?**

Sketch the graph of y = -2|x - 1| + 2 x y 1 -1 1 2 3 -2 Place 1 in the middle of the table. Choose 2 values less and 2 values more, evenly spaced. 2 -2 When there’s a negative coefficient before the absolute value symbol, the graph will open down.

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**What value of x will make the expression inside the absolute value sign equal to 0?**

Sketch the graph of x y -2 -4 -3 -2 -1 -2 Place -2 in the middle of the table. Choose 2 values less and 2 values more, evenly spaced. -2.5 -3 -2.5 -2 When there’s a positive coefficient before the absolute value symbol, the graph will open up.

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**Is there a way to easily tell what the y value of the vertex will be?**

y = |x| + 1 What will the x value of the vertex be? 1 If x is 0, what is y? y = |x - 2| - 5 What will the x value of the vertex be? 2 -5 If x is 2, what is y? y = |x + 3| - 4 -3 What will the x value of the vertex be? -4 If x is -3, what is y? 1 y = 2|x - 1| + 7 What will the x value of the vertex be? 7 If x is 1, what is y?

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**What will be the coordinates of the vertex?**

(0, 3) (-8, 0) (0, -5) (-9, -14) (-2, 0) (2, 6) (1, 5) y = |x| + 3 y = |x + 8| y = |x| - 5 y = |x + 9| - 14 y = -5|x + 2| y = 2|2x – 4| + 6 y = -|x – 1| + 5

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