{"id":63,"date":"2025-02-13T22:43:20","date_gmt":"2025-02-13T22:43:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-linear-functions\/"},"modified":"2026-01-09T20:54:25","modified_gmt":"2026-01-09T20:54:25","slug":"graphs-of-linear-functions","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-linear-functions\/","title":{"raw":"Graphs of Linear Functions: Learn It 5","rendered":"Graphs of Linear Functions: Learn It 5"},"content":{"raw":"

Finding the [latex]x[\/latex]-intercept of a Line<\/h2>\r\nSo far we have only discussed the [latex]y-[\/latex]intercepts of functions: the point at which the graph of a function crosses the [latex]y[\/latex]-axis. A function may also have an [latex]x[\/latex]-intercept,<\/strong> which is the [latex]x[\/latex]-coordinate of the point where the graph of a function crosses the [latex]x[\/latex]-axis. In other words, it is the input value when the output value is zero.\r\n\r\nTo find the [latex]x[\/latex]-intercept, set the function [latex]f[\/latex]([latex]x[\/latex]) equal to zero and solve for the value of [latex]x[\/latex]. For example, consider the function shown:\r\n

[latex]f\\left(x\\right)=3x - 6[\/latex]<\/p>\r\nSet the function equal to [latex]0[\/latex] and solve for [latex]x[\/latex].\r\n

[latex]\\begin{array}{l}0=3x - 6\\hfill \\\\ 6=3x\\hfill \\\\ 2=x\\hfill \\\\ x=2\\hfill \\end{array}[\/latex]<\/p>\r\nThe graph of the function crosses the [latex]x[\/latex]-axis at the point [latex](2, 0)[\/latex].\r\n\r\n

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[latex]x[\/latex]-intercept of a Line<\/h3>\r\nThe [latex]x[\/latex]-intercept<\/strong> of a function is the value of [latex]x[\/latex] where [latex]f(x) = 0[\/latex]. It can be found by solving the equation [latex]0 = mx + b[\/latex].\r\n\r\n<\/div>\r\n<\/section>
Do all linear functions have [latex]x[\/latex]-intercepts?<\/strong>\r\n\r\n
\r\n\r\nNo. However, linear functions of the form [latex]y = c[\/latex], where [latex]c[\/latex] is a nonzero real number are the only examples of linear functions with no [latex]x[\/latex]-intercept. For example, [latex]y = 5[\/latex] is a horizontal line [latex]5[\/latex] units above the [latex]x[\/latex]-axis. This function has no [latex]x[\/latex]-intercepts.\r\n\r\n
\"Graph<\/center><\/section>
Find the [latex]x[\/latex]-intercept of the following:\r\n
[latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex]<\/center>\r\n[reveal-answer q=\"400055\"]Show Solution[\/reveal-answer] [hidden-answer a=\"400055\"] Set the function equal to zero to solve for [latex]x[\/latex].\r\n

[latex]\\begin{array}{l}0=\\frac{1}{2}x - 3\\\\ 3=\\frac{1}{2}x\\\\ 6=x\\\\ x=6\\end{array}[\/latex]<\/p>\r\nThe graph crosses the [latex]x[\/latex]-axis at the point [latex](6, 0)[\/latex].\r\n[latex]\\\\[\/latex]\r\nAnalysis of the Solution<\/strong> [latex]\\\\[\/latex]A graph of the function is shown below. We can see that the [latex]x[\/latex]-intercept is [latex](6, 0)[\/latex] as expected.\r\n\r\n

\"The<\/center>
The graph of the linear function [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex]. \u00a0[\/hidden-answer]<\/span><\/strong><\/center><\/section>
[ohm_question hide_question_numbers=1]318707[\/ohm_question]<\/section>","rendered":"

Finding the [latex]x[\/latex]-intercept of a Line<\/h2>\n

So far we have only discussed the [latex]y-[\/latex]intercepts of functions: the point at which the graph of a function crosses the [latex]y[\/latex]-axis. A function may also have an [latex]x[\/latex]-intercept,<\/strong> which is the [latex]x[\/latex]-coordinate of the point where the graph of a function crosses the [latex]x[\/latex]-axis. In other words, it is the input value when the output value is zero.<\/p>\n

To find the [latex]x[\/latex]-intercept, set the function [latex]f[\/latex]([latex]x[\/latex]) equal to zero and solve for the value of [latex]x[\/latex]. For example, consider the function shown:<\/p>\n

[latex]f\\left(x\\right)=3x - 6[\/latex]<\/p>\n

Set the function equal to [latex]0[\/latex] and solve for [latex]x[\/latex].<\/p>\n

[latex]\\begin{array}{l}0=3x - 6\\hfill \\\\ 6=3x\\hfill \\\\ 2=x\\hfill \\\\ x=2\\hfill \\end{array}[\/latex]<\/p>\n

The graph of the function crosses the [latex]x[\/latex]-axis at the point [latex](2, 0)[\/latex].<\/p>\n

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\n

[latex]x[\/latex]-intercept of a Line<\/h3>\n

The [latex]x[\/latex]-intercept<\/strong> of a function is the value of [latex]x[\/latex] where [latex]f(x) = 0[\/latex]. It can be found by solving the equation [latex]0 = mx + b[\/latex].<\/p>\n<\/div>\n<\/section>\n

Do all linear functions have [latex]x[\/latex]-intercepts?<\/strong><\/p>\n
\n

No. However, linear functions of the form [latex]y = c[\/latex], where [latex]c[\/latex] is a nonzero real number are the only examples of linear functions with no [latex]x[\/latex]-intercept. For example, [latex]y = 5[\/latex] is a horizontal line [latex]5[\/latex] units above the [latex]x[\/latex]-axis. This function has no [latex]x[\/latex]-intercepts.<\/p>\n

\"Graph<\/div>\n<\/section>\n
Find the [latex]x[\/latex]-intercept of the following:<\/p>\n
[latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex]<\/div>\n