{"id":4018,"date":"2024-09-16T17:49:05","date_gmt":"2024-09-16T17:49:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=4018"},"modified":"2025-08-13T23:32:25","modified_gmt":"2025-08-13T23:32:25","slug":"graphs-of-linear-functions-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/graphs-of-linear-functions-learn-it-5\/","title":{"raw":"Graphs of Linear Functions: Learn It 5","rendered":"Graphs of Linear Functions: Learn It 5"},"content":{"raw":"

Determining Whether Lines are Parallel or Perpendicular<\/h2>\r\nThe two lines in the graph below\u00a0are parallel lines<\/strong>: they will never intersect. Notice that they have exactly the same steepness which means their slopes are identical. The only difference between the two lines is the y<\/em>-intercept. If we shifted one line vertically toward the y<\/em>-intercept of the other, they would become the same line.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Parallel lines.[\/caption]\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"535\"]\"The Comparison of functions and their slopes[\/caption]\r\n\r\nWe can determine from their equations whether two lines are parallel by comparing their slopes. If the slopes are the same and the [latex]y[\/latex]-intercepts are different, the lines are parallel. If the slopes are different, the lines are not parallel.\r\n\r\nUnlike parallel lines, perpendicular lines<\/strong> do intersect. Their intersection forms a right or[latex] 90[\/latex]-degree angle. The two lines below are perpendicular.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Perpendicular lines.[\/caption]\r\n\r\nPerpendicular lines do not have the same slope. The slopes of perpendicular lines are different from one another in a specific way. The slope of one line is the negative reciprocal of the slope of the other line.\r\n\r\n
The product of a number and its reciprocal is [latex]1[\/latex]. If [latex]{m}_{1}\\text{ and }{m}_{2}[\/latex] are negative reciprocals of one another, they can be multiplied together to yield [latex]-1[\/latex].\r\n

[latex]{m}_{1}*{m}_{2}=-1[\/latex]<\/p>\r\nTo find the reciprocal of a number, divide [latex]1[\/latex] by the number. So the reciprocal of [latex]8[\/latex] is [latex]\\frac{1}{8}[\/latex], and the reciprocal of [latex]\\frac{1}{8}[\/latex] is [latex]8[\/latex]. To find the negative reciprocal, first find the reciprocal and then change the sign.\r\n\r\n<\/section>As with parallel lines, we can determine whether two lines are perpendicular by comparing their slopes. The slope of each line below is the negative reciprocal of the other so the lines are perpendicular.\r\n

[latex]\\begin{array}{ll}f\\left(x\\right)=\\frac{1}{4}x+2\\hfill & \\text{negative reciprocal of }\\frac{1}{4}\\text{ is }-4\\hfill \\\\ f\\left(x\\right)=-4x+3\\hfill & \\text{negative reciprocal of }-4\\text{ is }\\frac{1}{4}\\hfill \\end{array}[\/latex]<\/p>\r\nThe product of the slopes is [latex]\u20131[\/latex].\r\n

[latex]-4\\left(\\frac{1}{4}\\right)=-1[\/latex]<\/p>\r\n\r\n

\r\n

parallel and perpendicular lines<\/h3>\r\nTwo lines are parallel lines<\/strong> if they do not intersect. The slopes of the lines are the same.\r\n\r\n
[latex]f\\left(x\\right)={m}_{1}x+{b}_{1}\\text{ and }g\\left(x\\right)={m}_{2}x+{b}_{2}\\text{ are parallel if }{m}_{1}={m}_{2}[\/latex].<\/center>\r\nIf and only if [latex]{b}_{1}={b}_{2}[\/latex] and [latex]{m}_{1}={m}_{2}[\/latex], we say the lines coincide. Coincident lines are the same line.\r\n[latex]\\\\[\/latex]\r\nTwo lines are perpendicular lines<\/strong> if they intersect at right angles.\r\n\r\n
[latex]f\\left(x\\right)={m}_{1}x+{b}_{1}\\text{ and }g\\left(x\\right)={m}_{2}x+{b}_{2}\\text{ are perpendicular if }{m}_{1}*{m}_{2}=-1,\\text{ and }{m}_{2}=-\\frac{1}{{m}_{1}}[\/latex].<\/center><\/section>
Given the functions below, identify the functions whose graphs are a pair of parallel lines and a pair of perpendicular lines.\r\n

[latex]\\begin{array}{l}f\\left(x\\right)=2x+3\\hfill & \\hfill & h\\left(x\\right)=-2x+2\\hfill \\\\ g\\left(x\\right)=\\frac{1}{2}x - 4\\hfill & \\hfill & j\\left(x\\right)=2x - 6\\hfill \\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"391905\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"391905\"]\r\n\r\nParallel lines have the same slope. Because the functions [latex]f\\left(x\\right)=2x+3[\/latex] and [latex]j\\left(x\\right)=2x - 6[\/latex] each have a slope of [latex]2[\/latex], they represent parallel lines.\r\n[latex]\\\\[\/latex]\r\nPerpendicular lines have negative reciprocal slopes. Because [latex]\u22122[\/latex] and [latex]\\frac{1}{2}[\/latex] are negative reciprocals, the equations, [latex]g\\left(x\\right)=\\frac{1}{2}x - 4[\/latex] and [latex]h\\left(x\\right)=-2x+2[\/latex] represent perpendicular lines.\r\n[latex]\\\\[\/latex]\r\nAnalysis of the Solution<\/strong>\r\n[latex]\\\\[\/latex]\r\nA graph of the lines is shown below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph The graph shows that the lines [latex]f\\left(x\\right)=2x+3[\/latex] and [latex]j\\left(x\\right)=2x - 6[\/latex] are parallel, and the lines [latex]g\\left(x\\right)=\\frac{1}{2}x - 4[\/latex] and [latex]h\\left(x\\right)=-2x+2[\/latex] are perpendicular.[\/caption][\/hidden-answer]<\/section>

[ohm2_question hide_question_numbers=1]20759[\/ohm2_question]<\/section>
[ohm2_question hide_question_numbers=1]20760[\/ohm2_question]<\/section>","rendered":"

Determining Whether Lines are Parallel or Perpendicular<\/h2>\n

The two lines in the graph below\u00a0are parallel lines<\/strong>: they will never intersect. Notice that they have exactly the same steepness which means their slopes are identical. The only difference between the two lines is the y<\/em>-intercept. If we shifted one line vertically toward the y<\/em>-intercept of the other, they would become the same line.<\/p>\n

\"Graph
Parallel lines.<\/figcaption><\/figure>\n
\"The
Comparison of functions and their slopes<\/figcaption><\/figure>\n

We can determine from their equations whether two lines are parallel by comparing their slopes. If the slopes are the same and the [latex]y[\/latex]-intercepts are different, the lines are parallel. If the slopes are different, the lines are not parallel.<\/p>\n

Unlike parallel lines, perpendicular lines<\/strong> do intersect. Their intersection forms a right or[latex]90[\/latex]-degree angle. The two lines below are perpendicular.<\/p>\n

\"Graph
Perpendicular lines.<\/figcaption><\/figure>\n

Perpendicular lines do not have the same slope. The slopes of perpendicular lines are different from one another in a specific way. The slope of one line is the negative reciprocal of the slope of the other line.<\/p>\n

The product of a number and its reciprocal is [latex]1[\/latex]. If [latex]{m}_{1}\\text{ and }{m}_{2}[\/latex] are negative reciprocals of one another, they can be multiplied together to yield [latex]-1[\/latex].<\/p>\n

[latex]{m}_{1}*{m}_{2}=-1[\/latex]<\/p>\n

To find the reciprocal of a number, divide [latex]1[\/latex] by the number. So the reciprocal of [latex]8[\/latex] is [latex]\\frac{1}{8}[\/latex], and the reciprocal of [latex]\\frac{1}{8}[\/latex] is [latex]8[\/latex]. To find the negative reciprocal, first find the reciprocal and then change the sign.<\/p>\n<\/section>\n

As with parallel lines, we can determine whether two lines are perpendicular by comparing their slopes. The slope of each line below is the negative reciprocal of the other so the lines are perpendicular.<\/p>\n

[latex]\\begin{array}{ll}f\\left(x\\right)=\\frac{1}{4}x+2\\hfill & \\text{negative reciprocal of }\\frac{1}{4}\\text{ is }-4\\hfill \\\\ f\\left(x\\right)=-4x+3\\hfill & \\text{negative reciprocal of }-4\\text{ is }\\frac{1}{4}\\hfill \\end{array}[\/latex]<\/p>\n

The product of the slopes is [latex]\u20131[\/latex].<\/p>\n

[latex]-4\\left(\\frac{1}{4}\\right)=-1[\/latex]<\/p>\n

\n

parallel and perpendicular lines<\/h3>\n

Two lines are parallel lines<\/strong> if they do not intersect. The slopes of the lines are the same.<\/p>\n

[latex]f\\left(x\\right)={m}_{1}x+{b}_{1}\\text{ and }g\\left(x\\right)={m}_{2}x+{b}_{2}\\text{ are parallel if }{m}_{1}={m}_{2}[\/latex].<\/div>\n

If and only if [latex]{b}_{1}={b}_{2}[\/latex] and [latex]{m}_{1}={m}_{2}[\/latex], we say the lines coincide. Coincident lines are the same line.
\n[latex]\\\\[\/latex]
\nTwo lines are perpendicular lines<\/strong> if they intersect at right angles.<\/p>\n

[latex]f\\left(x\\right)={m}_{1}x+{b}_{1}\\text{ and }g\\left(x\\right)={m}_{2}x+{b}_{2}\\text{ are perpendicular if }{m}_{1}*{m}_{2}=-1,\\text{ and }{m}_{2}=-\\frac{1}{{m}_{1}}[\/latex].<\/div>\n<\/section>\n
Given the functions below, identify the functions whose graphs are a pair of parallel lines and a pair of perpendicular lines.<\/p>\n

[latex]\\begin{array}{l}f\\left(x\\right)=2x+3\\hfill & \\hfill & h\\left(x\\right)=-2x+2\\hfill \\\\ g\\left(x\\right)=\\frac{1}{2}x - 4\\hfill & \\hfill & j\\left(x\\right)=2x - 6\\hfill \\end{array}[\/latex]<\/p>\n