{"id":265,"date":"2023-06-21T13:22:50","date_gmt":"2023-06-21T13:22:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-graphs-of-exponential-functions\/"},"modified":"2023-06-21T13:22:50","modified_gmt":"2023-06-21T13:22:50","slug":"summary-graphs-of-exponential-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-graphs-of-exponential-functions\/","title":{"raw":"Summary: Graphs of Exponential Functions","rendered":"Summary: Graphs of Exponential Functions"},"content":{"raw":"\n\n
Key Equations<\/h2>\n
\n\n
\n
General Form for the Transformation of the Parent Function [latex]\\text{ }f\\left(x\\right)={b}^{x}[\/latex]<\/td>\n
The graph of the function [latex]f\\left(x\\right)={b}^{x}[\/latex] has a y-<\/em>intercept at [latex]\\left(0, 1\\right)[\/latex], domain of [latex]\\left(-\\infty , \\infty \\right)[\/latex], range of [latex]\\left(0, \\infty \\right)[\/latex], and horizontal asymptote of [latex]y=0[\/latex].<\/li>\n \t
If [latex]b>1[\/latex], the function is increasing. The left tail of the graph will approach the asymptote [latex]y=0[\/latex], and the right tail will increase without bound.<\/li>\n \t
If 0 < b<\/em> < 1, the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote [latex]y=0[\/latex].<\/li>\n \t
The equation [latex]f\\left(x\\right)={b}^{x}+d[\/latex] represents a vertical shift of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\n \t
The equation [latex]f\\left(x\\right)={b}^{x+c}[\/latex] represents a horizontal shift of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\n \t
The equation [latex]f\\left(x\\right)=a{b}^{x}[\/latex], where [latex]a>0[\/latex], represents a vertical stretch if [latex]|a|>1[\/latex] or compression if [latex]0<|a|<1[\/latex] of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\n \t
When the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] is multiplied by \u20131, the result, [latex]f\\left(x\\right)=-{b}^{x}[\/latex], is a reflection about the x<\/em>-axis. When the input is multiplied by \u20131, the result, [latex]f\\left(x\\right)={b}^{-x}[\/latex], is a reflection about the y<\/em>-axis.<\/li>\n \t
All transformations of the exponential function can be summarized by the general equation [latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex].<\/li>\n \t
Using the general equation [latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex], we can write the equation of a function given its description.<\/li>\n \t
Approximate solutions of the equation [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex] can be found using a graphing calculator.<\/li>\n<\/ul>\n\n","rendered":"
Key Equations<\/h2>\n
\n\n
\n
General Form for the Transformation of the Parent Function [latex]\\text{ }f\\left(x\\right)={b}^{x}[\/latex]<\/td>\n
The graph of the function [latex]f\\left(x\\right)={b}^{x}[\/latex] has a y-<\/em>intercept at [latex]\\left(0, 1\\right)[\/latex], domain of [latex]\\left(-\\infty , \\infty \\right)[\/latex], range of [latex]\\left(0, \\infty \\right)[\/latex], and horizontal asymptote of [latex]y=0[\/latex].<\/li>\n
If [latex]b>1[\/latex], the function is increasing. The left tail of the graph will approach the asymptote [latex]y=0[\/latex], and the right tail will increase without bound.<\/li>\n
If 0 < b<\/em> < 1, the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote [latex]y=0[\/latex].<\/li>\n
The equation [latex]f\\left(x\\right)={b}^{x}+d[\/latex] represents a vertical shift of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\n
The equation [latex]f\\left(x\\right)={b}^{x+c}[\/latex] represents a horizontal shift of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\n
The equation [latex]f\\left(x\\right)=a{b}^{x}[\/latex], where [latex]a>0[\/latex], represents a vertical stretch if [latex]|a|>1[\/latex] or compression if [latex]0<|a|<1[\/latex] of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\n
When the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] is multiplied by \u20131, the result, [latex]f\\left(x\\right)=-{b}^{x}[\/latex], is a reflection about the x<\/em>-axis. When the input is multiplied by \u20131, the result, [latex]f\\left(x\\right)={b}^{-x}[\/latex], is a reflection about the y<\/em>-axis.<\/li>\n
All transformations of the exponential function can be summarized by the general equation [latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex].<\/li>\n
Using the general equation [latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex], we can write the equation of a function given its description.<\/li>\n
Approximate solutions of the equation [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex] can be found using a graphing calculator.<\/li>\n<\/ul>\n\n\t\t\t \n\t\t\t