{"id":842,"date":"2025-07-15T20:52:14","date_gmt":"2025-07-15T20:52:14","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=842"},"modified":"2026-01-12T20:51:11","modified_gmt":"2026-01-12T20:51:11","slug":"quadratic-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/quadratic-functions-learn-it-3\/","title":{"raw":"Quadratic Functions: Learn It 3","rendered":"Quadratic Functions: Learn It 3"},"content":{"raw":"

Transformations of Quadratic Functions Standard Form<\/h2>\r\nThe standard form is useful for determining how the graph is transformed <\/strong>from the graph of [latex]y={x}^{2}[\/latex].\r\n\r\n
The standard form of a quadratic function<\/strong> presents the function in the form\r\n

[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\r\nwhere [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function<\/strong>.\r\n\r\n<\/section>\r\n

Shift Up and Down by Changing the Value of [latex]k[\/latex]<\/h3>\r\n
\r\n

vertical shift of a parabola<\/h3>\r\nYou can represent a vertical (up, down) shift<\/strong> of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, [latex]k[\/latex].\r\n\r\n \r\n

[latex]f(x)=x^2 + k[\/latex]<\/p>\r\n \r\n\r\nIf [latex]k>0[\/latex], the graph shifts upward, whereas if [latex]k<0[\/latex], the graph shifts downward.\r\n\r\n<\/section>

Use an online graphing calculator to plot the function [latex]f(x)=x^2+k[\/latex].Now change the [latex]k[\/latex] value to shift the graph down [latex]4[\/latex] units, then up [latex]4[\/latex] units.What are the equations for each transformation?[reveal-answer q=\"941360\"]Show Solution[\/reveal-answer][hidden-answer a=\"941360\"]The equation for the graph of [latex]f(x)=x^2[\/latex] that has been shifted up [latex]4[\/latex] units is:\r\n

[latex]f(x)=x^2+4[\/latex]<\/p>\r\nThe equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted down [latex]4[\/latex] units is\r\n

[latex]f(x)=x^2-4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n

Shift left and right by changing the value of [latex]h[\/latex]<\/h3>\r\n
\r\n

horizontal shift of a parabola<\/h3>\r\nYou can represent a horizontal (left, right) shift<\/strong> of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, [latex]h[\/latex], to the variable [latex]x[\/latex], before squaring.\r\n\r\n \r\n

[latex]f(x)=(x-h)^2 [\/latex]<\/p>\r\n \r\n\r\nIf [latex]h>0[\/latex], the graph shifts toward the right and if [latex]h<0[\/latex], the graph shifts to the left.\r\n\r\n<\/section>

Remember that the negative sign inside the argument of the vertex form of a parabola (in the parentheses with the variable [latex]x[\/latex] ) is part of the formula [latex]f(x)=(x-h)^2 +k[\/latex].\r\n[latex]\\\\[\/latex]\r\nIf [latex]h>0[\/latex], we have\u00a0[latex]f(x)=(x-h)^2 +k[\/latex]. You'll see the negative sign, but the graph will shift right.\r\n[latex]\\\\[\/latex]\r\nIf\u00a0 [latex]h<0[\/latex], we have\u00a0[latex]f(x)=(x-(-h))^2 +k \\rightarrow f(x)=(x+h)^2+k[\/latex]. You'll see the positive sign, but the graph will shift left.<\/section>
Use an online graphing calculator to plot the function [latex]f(x)=(x-h)^2[\/latex].Now change the [latex]h[\/latex] value to shift the graph [latex]2[\/latex] units to the right, then [latex]2[\/latex] units to the left.What are the equations for each transformation?\r\n\r\n[reveal-answer q=\"978434\"]Show Solution[\/reveal-answer][hidden-answer a=\"978434\"]The equation for the graph of [latex]f(x)=x^2[\/latex] that has been shifted right [latex]2[\/latex] units is:\r\n

[latex]f(x)=(x-2)^2[\/latex]<\/p>\r\nThe equation for the graph of\u00a0[latex]f(x)=^2[\/latex] that has been shifted left [latex]2[\/latex] units is\r\n

[latex]f(x)=(x+2)^2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><\/section>\r\n

Stretch or compress by changing the value of [latex]a[\/latex].<\/h3>\r\n
\r\n

stretch or compress a parabola<\/h3>\r\nYou can represent a stretch<\/strong> or compression<\/strong> (narrowing, widening)\u00a0of the graph of [latex]f(x)=x^2[\/latex] by\u00a0multiplying the squared variable by a constant, [latex]a[\/latex].\r\n\r\n \r\n

[latex]f(x)=ax^2 [\/latex]<\/p>\r\n \r\n\r\nThe magnitude of [latex]a[\/latex]\u00a0indicates the stretch of the graph.\r\n\r\n \r\n\r\nIf [latex]|a|>1[\/latex], the point associated with a particular [latex]x[\/latex]-value shifts farther from the [latex]x[\/latex]-<\/em>axis, so the graph appears to become narrower, and there is a vertical stretch<\/strong>.\r\n\r\n \r\n\r\nBut if [latex]|a|<1[\/latex], the point associated with a particular [latex]x[\/latex]-value shifts closer to the [latex]x[\/latex]-<\/em>axis, so the graph appears to become wider, but in fact there is a vertical compression<\/strong>.\r\n\r\n<\/section>

Use an online graphing calculator to plot the function [latex]f(x)=ax^2[\/latex].Now adjust the [latex]a[\/latex] value to create a graph that has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex] and another that has been vertically stretched by a factor of [latex]3[\/latex]. [latex]\\\\[\/latex]\r\n\r\nWhat are the equations of the two graphs?[reveal-answer q=\"391411\"]Show Solution[\/reveal-answer][hidden-answer a=\"391411\"]The equation for the graph of [latex]f(x)=x^2[\/latex] that has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex] is\r\n

[latex]f(x)=\\frac{1}{2}x^2[\/latex]<\/p>\r\nThe equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been vertically stretched by a factor of [latex]3[\/latex] is\r\n

[latex]f(x)=3x^2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

[ohm_question height=\"400\" hide_question_numbers=1]318787[\/ohm_question]<\/section>