Key Equations
General Form for the Translation of the Parent Function f(x)=bx f(x)=bx | f(x)=abx+c+df(x)=abx+c+d |
Key Concepts
- The graph of the function f(x)=bxf(x)=bx has a y-intercept at (0,1)(0,1), domain (−∞,∞)(−∞,∞), range (0,∞)(0,∞), and horizontal asymptote y=0y=0.
- If b>1b>1, the function is increasing. The left tail of the graph will approach the asymptote y=0y=0, and the right tail will increase without bound.
- If 0 < b < 1, the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote y=0y=0.
- The equation f(x)=bx+df(x)=bx+d represents a vertical shift of the parent function f(x)=bxf(x)=bx.
- The equation f(x)=bx+cf(x)=bx+c represents a horizontal shift of the parent function f(x)=bxf(x)=bx.
- Approximate solutions of the equation f(x)=bx+c+df(x)=bx+c+d can be found using a graphing calculator.
- The equation f(x)=abxf(x)=abx, where a>0a>0, represents a vertical stretch if |a|>1|a|>1 or compression if 0<|a|<10<|a|<1 of the parent function f(x)=bxf(x)=bx.
- When the parent function f(x)=bxf(x)=bx is multiplied by –1, the result, f(x)=−bxf(x)=−bx, is a reflection about the x-axis. When the input is multiplied by –1, the result, f(x)=b−xf(x)=b−x, is a reflection about the y-axis.
- All translations of the exponential function can be summarized by the general equation f(x)=abx+c+df(x)=abx+c+d.
- Using the general equation f(x)=abx+c+df(x)=abx+c+d, we can write the equation of a function given its description.
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- Precalculus. Authored by: Jay Abramson, et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175. License: CC BY: Attribution. License Terms: Download For Free at : http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.
- College Algebra. Authored by: Abramson, Jay et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2. License: CC BY: Attribution. License Terms: Download for free at http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2