{"id":1893,"date":"2024-06-24T18:58:36","date_gmt":"2024-06-24T18:58:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1893"},"modified":"2025-08-15T02:16:25","modified_gmt":"2025-08-15T02:16:25","slug":"introduction-to-power-and-polynomial-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-power-and-polynomial-functions-learn-it-2\/","title":{"raw":"Introduction to Power and Polynomial Functions: Learn It 2","rendered":"Introduction to Power and Polynomial Functions: Learn It 2"},"content":{"raw":"

Identifying End Behavior of Power Functions<\/h2>\r\n[caption id=\"attachment_4236\" align=\"alignright\" width=\"419\"]\"a Graph demonstrating x to even powers[\/caption]\r\n\r\nThe behavior of the graph of a function as the input values get very small ( [latex]x\\to -\\infty[\/latex] ) and get very large ( [latex]x\\to \\infty[\/latex] ) is referred to as the end behavior<\/strong> of the function. We can use words or symbols to describe end behavior. Let's look at the graph of power functions with a non-negative integer exponent that has the form [latex]f(x) = ax^n[\/latex] where [latex]n[\/latex] is a non-negative integer ([latex]0, 1, 2, 3, ...[\/latex]).\r\n

Even-Power Functions<\/h3>\r\nThe graph shows the graphs of [latex]h\\left(x\\right)={x}^{2},g\\left(x\\right)={x}^{4}[\/latex], and [latex]f\\left(x\\right)={x}^{6}[\/latex], which are all power functions with even, whole-number powers. Notice that these\r\n\r\ngraphs have similar shapes, very much like that of the quadratic function. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.\r\n\r\nTo describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol [latex]\\infty[\/latex] for positive infinity and [latex]-\\infty[\/latex] for negative infinity. When we say that \"[latex]x[\/latex] approaches infinity,\" which can be symbolically written as [latex]x\\to \\infty[\/latex], we are describing a behavior; we are saying that [latex]x[\/latex] is increasing without bound.\r\n\r\n
\r\n

even-power functions<\/h3>\r\nWith even-powered power functions, as the input increases or decreases without bound, the output values become very large, positive numbers.\r\n\r\n \r\n\r\nEquivalently, we could describe this behavior by saying that as [latex]x[\/latex] approaches positive or negative infinity, the [latex]f(x)[\/latex] values increase without bound.\r\n

[latex]\\text{as }x\\to \\pm \\infty , f(x) \\to \\infty[\/latex]<\/p>\r\n\r\n<\/section>

[ohm2_question hide_question_numbers=1]24599[\/ohm2_question]<\/section>\r\n\r\n[caption id=\"attachment_4237\" align=\"alignright\" width=\"373\"]\"a Graph demonstrating x to odd powers[\/caption]\r\n

Odd-Power Functions<\/h3>\r\nThe graph shows [latex]f\\left(x\\right)={x}^{3},g\\left(x\\right)={x}^{5}, \\text{ and } h\\left(x\\right)={x}^{7}[\/latex], which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function. As the power increases, the graphs flatten near the origin and become steeper away from the origin.\r\n\r\nThese examples illustrate that functions of the form [latex]f(x)={x}^{n}[\/latex] reveal symmetry of one kind or another. First, in the even-powered power functions, we see that even functions of the form [latex]f(x)={x}^{n}\\text{, }n\\text{ even,}[\/latex] are symmetric about the y<\/em>-axis. In the odd-powered power functions, we see that odd functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n\\text{ odd,}[\/latex] are symmetric about the origin.\r\n\r\n
\r\n

odd-power functions<\/h3>\r\nFor these odd power functions, as [latex]x[\/latex] approaches negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound. As [latex]x[\/latex] approaches positive infinity, [latex]f(x)[\/latex] increases without bound.\r\n\r\n \r\n

[latex]\\begin{array}{c}\\text{as } x\\to -\\infty , f(x) \\to -\\infty \\\\ \\text{as } x\\to \\infty , f(x) \\to \\infty \\end{array}[\/latex]<\/p>\r\n\r\n<\/section>

The table\u00a0below shows the end behavior of power functions of the form [latex]f\\left(x\\right)=a{x}^{n}[\/latex] where [latex]n[\/latex] is a non-negative integer depending on the power and the constant.\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n
<\/th>\r\nEven power<\/th>\r\nOdd power<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
\r\n

Positive constant <\/strong><\/p>\r\n

a<\/i> > 0<\/strong><\/p>\r\n<\/td>\r\n

\r\n\r\n[caption id=\"attachment_4485\" align=\"alignnone\" width=\"356\"]\"Graph Graph of an even-powered function[\/caption]<\/td>\r\n\r\n\r\n[caption id=\"attachment_4487\" align=\"alignnone\" width=\"359\"]\"Graph Graph of an odd-powered function[\/caption]<\/td>\r\n<\/tr>\r\n
Negative constant<\/strong>\r\n\r\na<\/i> < 0<\/strong><\/td>\r\n\r\n\r\n[caption id=\"attachment_4488\" align=\"alignnone\" width=\"375\"]\"Graph Graph of an even-powered function with a negative constant[\/caption]<\/td>\r\n\r\n\r\n[caption id=\"attachment_4489\" align=\"aligncenter\" width=\"342\"]\"Graph Graph of an odd-powered function with a negative constant[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n<\/center><\/section>
Describe the end behavior of the graph of [latex]f\\left(x\\right)={x}^{8}[\/latex].[reveal-answer q=\"556064\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"556064\"]The coefficient is [latex]1[\/latex] (positive) and the exponent of the power function is [latex]8[\/latex] (an even number). As [latex]x[\/latex](input)\u00a0approaches infinity, [latex]f\\left(x\\right)[\/latex] (output) increases without bound. We write as [latex]x\\to \\infty , f\\left(x\\right)\\to \\infty [\/latex]. As [latex]x[\/latex]\u00a0approaches negative infinity, the output increases without bound. In symbolic form, as [latex]x\\to -\\infty , f\\left(x\\right)\\to \\infty[\/latex]. We can graphically represent the function.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of f(x)=x^8[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
[ohm2_question hide_question_numbers=1]24600[\/ohm2_question]<\/section>","rendered":"

Identifying End Behavior of Power Functions<\/h2>\n
\"a
Graph demonstrating x to even powers<\/figcaption><\/figure>\n

The behavior of the graph of a function as the input values get very small ( [latex]x\\to -\\infty[\/latex] ) and get very large ( [latex]x\\to \\infty[\/latex] ) is referred to as the end behavior<\/strong> of the function. We can use words or symbols to describe end behavior. Let’s look at the graph of power functions with a non-negative integer exponent that has the form [latex]f(x) = ax^n[\/latex] where [latex]n[\/latex] is a non-negative integer ([latex]0, 1, 2, 3, ...[\/latex]).<\/p>\n

Even-Power Functions<\/h3>\n

The graph shows the graphs of [latex]h\\left(x\\right)={x}^{2},g\\left(x\\right)={x}^{4}[\/latex], and [latex]f\\left(x\\right)={x}^{6}[\/latex], which are all power functions with even, whole-number powers. Notice that these<\/p>\n

graphs have similar shapes, very much like that of the quadratic function. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.<\/p>\n

To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol [latex]\\infty[\/latex] for positive infinity and [latex]-\\infty[\/latex] for negative infinity. When we say that “[latex]x[\/latex] approaches infinity,” which can be symbolically written as [latex]x\\to \\infty[\/latex], we are describing a behavior; we are saying that [latex]x[\/latex] is increasing without bound.<\/p>\n

\n

even-power functions<\/h3>\n

With even-powered power functions, as the input increases or decreases without bound, the output values become very large, positive numbers.<\/p>\n

 <\/p>\n

Equivalently, we could describe this behavior by saying that as [latex]x[\/latex] approaches positive or negative infinity, the [latex]f(x)[\/latex] values increase without bound.<\/p>\n

[latex]\\text{as }x\\to \\pm \\infty , f(x) \\to \\infty[\/latex]<\/p>\n<\/section>\n