{"id":825,"date":"2025-07-15T20:08:57","date_gmt":"2025-07-15T20:08:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=825"},"modified":"2025-12-30T19:23:59","modified_gmt":"2025-12-30T19:23:59","slug":"absolute-value-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/absolute-value-functions-learn-it-2\/","title":{"raw":"Absolute Value Functions: Learn It 2","rendered":"Absolute Value Functions: Learn It 2"},"content":{"raw":"

Graphing an Absolute Value Function<\/h2>\r\n

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>.\r\n<\/span><\/p>\r\n\"Graph\r\n\r\nThe graph of [latex]y=2\\left|x - 3\\right|+4[\/latex] is an absolute value functions after three transformations. The graph of [latex]y=|x|[\/latex] has been shifted right 3 units resulting in [latex]f(x)=\\left|x-3\\right|[\/latex], then vertically stretched by a factor of 2 ([latex]2\\left|x-3\\right|[\/latex]), and shifted up 4 units. This means that the corner point is located at [latex]\\left(3,4\\right)[\/latex] for this transformed function.\r\n\r\n\"Graph\r\n

\r\n
Write an equation for the function graphed.\"Graph[reveal-answer q=\"729255\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"729255\"]\r\n\r\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.\r\n\r\n\"Graph\r\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.\r\n<\/span><\/p>\r\n\"Graph\r\n

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

[latex]\\begin{align}&f\\left(x\\right)=2\\left|x - 3\\right|-2, && \\text{treating the stretch as a vertical stretch,} \\\\[2mm] \\text{or } &f\\left(x\\right)=\\left|2\\left(x - 3\\right)\\right|-2, && \\text{treating the stretch as a horizontal compression}. \\end{align}[\/latex]<\/p>\r\n\r\n

Analysis of the Solution<\/h4>\r\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>\r\n[\/hidden-answer]\r\n\r\n<\/section>

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].[latex]f\\left(x\\right)=a|x - 3|-2[\/latex]Now substituting in the point (1, 2)\r\n\r\n[latex]\\begin{align}&2=a|1 - 3|-2 \\\\ &4=2a \\\\ &a=2 \\end{align}[\/latex]\r\n\r\n<\/section><\/div>\r\n<\/div>\r\n
[ohm_question hide_question_numbers=1]317635[\/ohm_question]\r\n\r\n<\/section><\/div>","rendered":"

Graphing an Absolute Value Function<\/h2>\n

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

The graph of [latex]y=2\\left|x - 3\\right|+4[\/latex] is an absolute value functions after three transformations. The graph of [latex]y=|x|[\/latex] has been shifted right 3 units resulting in [latex]f(x)=\\left|x-3\\right|[\/latex], then vertically stretched by a factor of 2 ([latex]2\\left|x-3\\right|[\/latex]), 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

\n
\n
Write an equation for the function graphed.\"Graph<\/p>\n