{"id":922,"date":"2025-07-16T18:20:54","date_gmt":"2025-07-16T18:20:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=922"},"modified":"2026-01-13T16:15:04","modified_gmt":"2026-01-13T16:15:04","slug":"graphs-of-polynomial-functions-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-polynomial-functions-learn-it-5\/","title":{"raw":"Graphs of Polynomial Functions: Learn It 5","rendered":"Graphs of Polynomial Functions: Learn It 5"},"content":{"raw":"

Using the Intermediate Value Theorem<\/h2>\r\nIn some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the [latex]x[\/latex]-axis, we can confirm that there is a zero between them. Consider a polynomial function [latex]f[\/latex]\u00a0whose graph is smooth and continuous.\r\n\r\n
\r\n

Intermediate Value Theorem<\/h3>\r\nThe Intermediate Value Theorem<\/strong> states that for two numbers [latex]a[\/latex]\u00a0and [latex]b[\/latex]\u00a0in the domain of [latex]f[\/latex],\u00a0if [latex]a\u00a0< b[\/latex]\u00a0and [latex]f\\left(a\\right)\\ne f\\left(b\\right)[\/latex], then the function [latex]f[\/latex]\u00a0takes on every value between [latex]f\\left(a\\right)[\/latex] and [latex]f\\left(b\\right)[\/latex].\r\n\r\n \r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"397\"]\"Graph The Intermediate Value Theorem can be used to show there exists a zero.[\/caption]\r\n\r\n<\/section>We can apply this theorem to a special case that is useful for graphing polynomial functions. If a point on the graph of a continuous function f<\/em>\u00a0at [latex]x=a[\/latex] lies above the [latex]x[\/latex]-axis and another point at [latex]x=b[\/latex] lies below the [latex]x[\/latex]-axis, there must exist a third point between [latex]x=a[\/latex] and [latex]x=b[\/latex] where the graph crosses the [latex]x[\/latex]-axis. Call this point [latex]\\left(c,\\text{ }f\\left(c\\right)\\right)[\/latex]. This means that we are assured there is a value\u00a0[latex]c[\/latex]\u00a0where [latex]f\\left(c\\right)=0[\/latex].\r\n\r\nIn other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the [latex]x[\/latex]-axis. The figure below\u00a0shows that there is a zero between [latex]a[\/latex]\u00a0and [latex]b[\/latex].\r\n\r\n
Show that the function [latex]f\\left(x\\right)={x}^{3}-5{x}^{2}+3x+6[\/latex]\u00a0has at least two real zeros between [latex]x=1[\/latex]\u00a0and [latex]x=4[\/latex].[reveal-answer q=\"33137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"33137\"]To start, evaluate [latex]f\\left(x\\right)[\/latex]\u00a0at the integer values [latex]x=1,2,3,\\text{ and }4[\/latex].\r\n<\/colgroup>\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/strong><\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]f(x)[\/latex]<\/strong><\/td>\r\n[latex]5[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]\u20133[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe see that one zero occurs at [latex]x=2[\/latex]. Also, since [latex]f\\left(3\\right)[\/latex] is negative and [latex]f\\left(4\\right)[\/latex] is positive, by the Intermediate Value Theorem, there must be at least one real zero between [latex]3[\/latex] and [latex]4[\/latex].\r\n\r\nWe have shown that there are at least two real zeros between [latex]x=1[\/latex]\u00a0and [latex]x=4[\/latex].\r\n\r\nAnalysis of the Solution<\/strong>\r\n\r\nWe can also graphically see that there are two real zeros between [latex]x=1[\/latex]\u00a0and [latex]x=4[\/latex].\r\n\r\n\"Graph\r\n[\/hidden-answer]\r\n\r\n<\/section>
[ohm_question hide_question_numbers=1]318829[\/ohm_question]<\/section>
[ohm_question hide_question_numbers=1]318830[\/ohm_question]<\/section>
[ohm_question hide_question_numbers=1]318831[\/ohm_question]<\/section>","rendered":"

Using the Intermediate Value Theorem<\/h2>\n

In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the [latex]x[\/latex]-axis, we can confirm that there is a zero between them. Consider a polynomial function [latex]f[\/latex]\u00a0whose graph is smooth and continuous.<\/p>\n

\n

Intermediate Value Theorem<\/h3>\n

The Intermediate Value Theorem<\/strong> states that for two numbers [latex]a[\/latex]\u00a0and [latex]b[\/latex]\u00a0in the domain of [latex]f[\/latex],\u00a0if [latex]a\u00a0< b[\/latex]\u00a0and [latex]f\\left(a\\right)\\ne f\\left(b\\right)[\/latex], then the function [latex]f[\/latex]\u00a0takes on every value between [latex]f\\left(a\\right)[\/latex] and [latex]f\\left(b\\right)[\/latex].\n\n \n\n\n\n

\"Graph
The Intermediate Value Theorem can be used to show there exists a zero.<\/figcaption><\/figure>\n<\/section>\n

We can apply this theorem to a special case that is useful for graphing polynomial functions. If a point on the graph of a continuous function f<\/em>\u00a0at [latex]x=a[\/latex] lies above the [latex]x[\/latex]-axis and another point at [latex]x=b[\/latex] lies below the [latex]x[\/latex]-axis, there must exist a third point between [latex]x=a[\/latex] and [latex]x=b[\/latex] where the graph crosses the [latex]x[\/latex]-axis. Call this point [latex]\\left(c,\\text{ }f\\left(c\\right)\\right)[\/latex]. This means that we are assured there is a value\u00a0[latex]c[\/latex]\u00a0where [latex]f\\left(c\\right)=0[\/latex].<\/p>\n

In other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the [latex]x[\/latex]-axis. The figure below\u00a0shows that there is a zero between [latex]a[\/latex]\u00a0and [latex]b[\/latex].<\/p>\n

Show that the function [latex]f\\left(x\\right)={x}^{3}-5{x}^{2}+3x+6[\/latex]\u00a0has at least two real zeros between [latex]x=1[\/latex]\u00a0and [latex]x=4[\/latex].<\/p>\n