{"id":1002,"date":"2015-11-12T18:35:32","date_gmt":"2015-11-12T18:35:32","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1002"},"modified":"2015-11-12T18:35:32","modified_gmt":"2015-11-12T18:35:32","slug":"graph-an-absolute-value-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/chapter\/graph-an-absolute-value-function\/","title":{"raw":"Graph an absolute value function","rendered":"Graph an absolute value function"},"content":{"raw":"

The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the origin<\/strong>.\n<\/span><\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"GraphFigure 3<\/b>[\/caption]\n\nFigure 4\u00a0is the graph of [latex]y=2\\left|x - 3\\right|+4\\\\[\/latex]. The graph of [latex]y=|x|\\\\[\/latex] has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at [latex]\\left(3,4\\right)\\\\[\/latex] for this transformed function.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"GraphFigure 4<\/b>[\/caption]\n\n

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Example 3: Writing an Equation for an Absolute Value Function<\/h3>\nWrite an equation for the function graphed in Figure 5.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"GraphFigure 5<\/b>[\/caption]\n\n<\/div>\n
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Solution<\/h3>\nThe basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"GraphFigure 6<\/b>[\/caption]\n

We also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. Instead, the width is equal to 1 times the vertical distance.\n<\/span><\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"GraphFigure 7<\/b>[\/caption]\n

From this information we can write the equation<\/p>\n\n

[latex]\\begin{cases}f\\left(x\\right)=2\\left|x - 3\\right|-2,\\hfill & \\text{treating the stretch as a vertical stretch, or}\\hfill \\\\ f\\left(x\\right)=\\left|2\\left(x - 3\\right)\\right|-2,\\hfill & \\text{treating the stretch as a horizontal compression}.\\hfill \\end{cases}\\\\[\/latex]<\/div>\n<\/div>\n
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Analysis of the Solution<\/h3>\n

Note that these equations are algebraically equivalent\u2014the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n

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Q & A<\/strong><\/p>\nIf we couldn\u2019t observe the stretch of the function from the graphs, could we algebraically determine it?<\/strong>\n

Yes. If we are unable to determine the stretch based on the width of the graph, we can solve for the stretch factor by putting in a known pair of values for [latex]x\\\\[\/latex] and [latex]f\\left(x\\right)\\\\[\/latex].<\/em><\/p>\n\n

[latex]f\\left(x\\right)=a|x - 3|-2\\\\[\/latex]<\/div>\n

Now substituting in the point <\/em>(1, 2)<\/p>\n\n

[latex]\\begin{cases}2=a|1 - 3|-2\\hfill \\\\ 4=2a\\hfill \\\\ a=2\\hfill \\end{cases}\\\\[\/latex]<\/div>\n<\/div>\n
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Try It 3<\/h3>\n

Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically flipped, and vertically shifted up 3 units.<\/p>\nSolution<\/a>\n\n<\/div>\n

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Q & A<\/strong><\/p>\nDo the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?\n<\/strong>\n

Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.\n<\/em><\/p>\n

No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points.<\/em><\/p>\n\n<\/div>\n

\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"GraphFigure 8.<\/b> (a) The absolute value function does not intersect the horizontal axis. (b) The absolute value function intersects the horizontal axis at one point. (c) The absolute value function intersects the horizontal axis at two points.[\/caption]\n\n<\/figure><\/section>
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The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the origin<\/strong>.
\n<\/span><\/p>\n

\"Graph<\/p>\n

Figure 3<\/b><\/p>\n<\/div>\n

Figure 4\u00a0is the graph of [latex]y=2\\left|x - 3\\right|+4\\\\[\/latex]. The graph of [latex]y=|x|\\\\[\/latex] has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at [latex]\\left(3,4\\right)\\\\[\/latex] for this transformed function.<\/p>\n

\"Graph<\/p>\n

Figure 4<\/b><\/p>\n<\/div>\n

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Example 3: Writing an Equation for an Absolute Value Function<\/h3>\n

Write an equation for the function graphed in Figure 5.<\/p>\n

\"Graph<\/p>\n

Figure 5<\/b><\/p>\n<\/div>\n<\/div>\n

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Solution<\/h3>\n

The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function.<\/p>\n

\"Graph<\/p>\n

Figure 6<\/b><\/p>\n<\/div>\n

We also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. Instead, the width is equal to 1 times the vertical distance.
\n<\/span><\/p>\n

\"Graph<\/p>\n

Figure 7<\/b><\/p>\n<\/div>\n

From this information we can write the equation<\/p>\n

[latex]\\begin{cases}f\\left(x\\right)=2\\left|x - 3\\right|-2,\\hfill & \\text{treating the stretch as a vertical stretch, or}\\hfill \\\\ f\\left(x\\right)=\\left|2\\left(x - 3\\right)\\right|-2,\\hfill & \\text{treating the stretch as a horizontal compression}.\\hfill \\end{cases}\\\\[\/latex]<\/div>\n<\/div>\n
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Analysis of the Solution<\/h3>\n

Note that these equations are algebraically equivalent\u2014the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

\n

Q & A<\/strong><\/p>\n

If we couldn\u2019t observe the stretch of the function from the graphs, could we algebraically determine it?<\/strong><\/p>\n

Yes. If we are unable to determine the stretch based on the width of the graph, we can solve for the stretch factor by putting in a known pair of values for [latex]x\\\\[\/latex] and [latex]f\\left(x\\right)\\\\[\/latex].<\/em><\/p>\n

[latex]f\\left(x\\right)=a|x - 3|-2\\\\[\/latex]<\/div>\n

Now substituting in the point <\/em>(1, 2)<\/p>\n

[latex]\\begin{cases}2=a|1 - 3|-2\\hfill \\\\ 4=2a\\hfill \\\\ a=2\\hfill \\end{cases}\\\\[\/latex]<\/div>\n<\/div>\n
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Try It 3<\/h3>\n

Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically flipped, and vertically shifted up 3 units.<\/p>\n

Solution<\/a><\/p>\n<\/div>\n

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Q & A<\/strong><\/p>\n

Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?
\n<\/strong><\/p>\n

Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.
\n<\/em><\/p>\n

No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points.<\/em><\/p>\n<\/div>\n

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\"Graph<\/p>\n

Figure 8.<\/b> (a) The absolute value function does not intersect the horizontal axis. (b) The absolute value function intersects the horizontal axis at one point. (c) The absolute value function intersects the horizontal axis at two points.<\/p>\n<\/div>\n<\/figure>\n<\/section>\n

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