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Study Guides > MATH 1314: College Algebra

Key Concepts

Key Equations

General Form for the Translation of the Parent Function  f(x)=bx\text{ }f\left(x\right)={b}^{x}\\ f(x)=abx+c+df\left(x\right)=a{b}^{x+c}+d\\

Key Concepts

  • The graph of the function f(x)=bxf\left(x\right)={b}^{x}\\ has a y-intercept at (0,1)\left(0, 1\right)\\, domain (,)\left(-\infty , \infty \right)\\, range (0,)\left(0, \infty \right)\\, 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\left(x\right)={b}^{x}+d\\ represents a vertical shift of the parent function f(x)=bxf\left(x\right)={b}^{x}\\.
  • The equation f(x)=bx+cf\left(x\right)={b}^{x+c}\\ represents a horizontal shift of the parent function f(x)=bxf\left(x\right)={b}^{x}\\.
  • Approximate solutions of the equation f(x)=bx+c+df\left(x\right)={b}^{x+c}+d\\ can be found using a graphing calculator.
  • The equation f(x)=abxf\left(x\right)=a{b}^{x}\\, 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\left(x\right)={b}^{x}\\.
  • When the parent function f(x)=bxf\left(x\right)={b}^{x}\\ is multiplied by –1, the result, f(x)=bxf\left(x\right)=-{b}^{x}\\, is a reflection about the x-axis. When the input is multiplied by –1, the result, f(x)=bxf\left(x\right)={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\left(x\right)=a{b}^{x+c}+d\\.
  • Using the general equation f(x)=abx+c+df\left(x\right)=a{b}^{x+c}+d\\, we can write the equation of a function given its description.

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