{"id":939,"date":"2024-04-09T16:41:13","date_gmt":"2024-04-09T16:41:13","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&p=939"},"modified":"2024-08-31T02:13:32","modified_gmt":"2024-08-31T02:13:32","slug":"basic-classes-of-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/basic-classes-of-functions-learn-it-2\/","title":{"raw":"Basic Classes of Functions: Learn It 2","rendered":"Basic Classes of Functions: Learn It 2"},"content":{"raw":"

Zeros of Polynomial Functions<\/h2>\r\n

To determine where a function [latex]f[\/latex] intersects the [latex]x[\/latex]-axis, we need to solve the equation [latex]f(x)=0[\/latex] for [latex]x[\/latex].<\/p>\r\n

In the case of the linear function [latex]f(x)=mx+b[\/latex], the [latex]x[\/latex]-intercept is given by solving the equation [latex]mx+b=0[\/latex]. Which can be found by [latex](\u2212\\frac{b}{m},0)[\/latex].<\/p>\r\n

In the case of a quadratic function, finding the [latex]x[\/latex]-intercept(s) requires finding the zeros of a quadratic equation: [latex]ax^2+bx+c=0[\/latex]. In some cases, it is easy to factor the polynomial [latex]ax^2+bx+c[\/latex] to find the zeros. If not, we make use of the quadratic formula.<\/p>\r\n

\r\n

The Quadratic Formula<\/h3>\r\n

Consider the quadratic equation<\/p>\r\n

[latex]ax^2+bx+c=0[\/latex],<\/div>\r\n

 <\/p>\r\n

where [latex]a\\ne 0[\/latex]. The solutions of this equation are given by the quadratic formula<\/p>\r\n

[latex]x=\\dfrac{\u2212b \\pm \\sqrt{b^2-4ac}}{2a}[\/latex]<\/div>\r\n

\u00a0<\/p>\r\n<\/section>\r\n

\r\n

The discriminant, given by, [latex]b^2-4ac[\/latex], determines the nature of a quadratic equation's solutions. A positive discriminant indicates two distinct real solutions, a discriminant of zero results in exactly one real solution, and a negative discriminant means the equation has no real solutions.<\/p>\r\n<\/section>\r\n

In the case of higher-degree polynomials, it may be more complicated to determine where the graph intersects the [latex]x[\/latex]-axis. For this content, we will only focus on finding the zeros of quadratic polynomials.<\/p>\r\n

\r\n

Consider the quadratic function [latex]f(x)=3x^2-6x+2[\/latex]. Find the zeros of [latex]f(x)[\/latex].<\/p>\r\n

[reveal-answer q=\"fs-id1170573352571\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"fs-id1170573352571\"]<\/p>\r\n\r\nTo find the zeros of the function [latex]f(x)=3x^2\u22126x+2[\/latex], we need to solve for when [latex]f(x)[\/latex] is equal to zero. This gives us the quadratic equation [latex]3x^2\u22126x+2=0[\/latex]. Using the quadratic formula:

[latex]\\begin{array}{l} x = \\frac{-(-6) \\pm \\sqrt{(-6)^2 - 4 \\cdot 3 \\cdot 2}}{2 \\cdot 3} \\\\ x = \\frac{6 \\pm \\sqrt{36 - 24}}{6} \\\\ x = \\frac{6 \\pm \\sqrt{12}}{6} \\\\ x = \\frac{6 \\pm 2\\sqrt{3}}{6} \\\\ x = 1 \\pm \\frac{\\sqrt{3}}{3} \\end{array}[\/latex]<\/center>Therefore, the zeros of the function are [latex]x=1\u2212\\frac{\\sqrt{3}}{3}[\/latex] and [latex]x=1+\\frac{\\sqrt{3}}{3}[\/latex].\r\n\r\n

[\/hidden-answer]<\/p>\r\n<\/section>\r\n

\r\n

[ohm_question hide_question_numbers=1]33476[\/ohm_question]<\/p>\r\n<\/section>","rendered":"

Zeros of Polynomial Functions<\/h2>\n

To determine where a function [latex]f[\/latex] intersects the [latex]x[\/latex]-axis, we need to solve the equation [latex]f(x)=0[\/latex] for [latex]x[\/latex].<\/p>\n

In the case of the linear function [latex]f(x)=mx+b[\/latex], the [latex]x[\/latex]-intercept is given by solving the equation [latex]mx+b=0[\/latex]. Which can be found by [latex](\u2212\\frac{b}{m},0)[\/latex].<\/p>\n

In the case of a quadratic function, finding the [latex]x[\/latex]-intercept(s) requires finding the zeros of a quadratic equation: [latex]ax^2+bx+c=0[\/latex]. In some cases, it is easy to factor the polynomial [latex]ax^2+bx+c[\/latex] to find the zeros. If not, we make use of the quadratic formula.<\/p>\n

\n

The Quadratic Formula<\/h3>\n

Consider the quadratic equation<\/p>\n

[latex]ax^2+bx+c=0[\/latex],<\/div>\n

 <\/p>\n

where [latex]a\\ne 0[\/latex]. The solutions of this equation are given by the quadratic formula<\/p>\n

[latex]x=\\dfrac{\u2212b \\pm \\sqrt{b^2-4ac}}{2a}[\/latex]<\/div>\n

\u00a0<\/p>\n<\/section>\n

\n

The discriminant, given by, [latex]b^2-4ac[\/latex], determines the nature of a quadratic equation’s solutions. A positive discriminant indicates two distinct real solutions, a discriminant of zero results in exactly one real solution, and a negative discriminant means the equation has no real solutions.<\/p>\n<\/section>\n

In the case of higher-degree polynomials, it may be more complicated to determine where the graph intersects the [latex]x[\/latex]-axis. For this content, we will only focus on finding the zeros of quadratic polynomials.<\/p>\n

\n

Consider the quadratic function [latex]f(x)=3x^2-6x+2[\/latex]. Find the zeros of [latex]f(x)[\/latex].<\/p>\n