Reciprocal Transformations (DP IB Analysis & Approaches (AA): HL): Revision Note

Reciprocal transformations

What is a reciprocal transformation?

• For the graph $y=f\left(x\right)$ the reciprocal transformation is $y=\frac{1}{f\left(x\right)}$

• It transforms points on the graph $y=f\left(x\right)$

  • by changing their $y$-coordinates

    • from a height of $y$ to a height of $\frac{1}{y}$

• All $x$-coordinates stay the same

  • Points that lie on the line y = 1 or the line y = -1 stay the same

How do I sketch a reciprocal transformation y = 1/f(x)?

• To sketch a reciprocal transformation, you need to know

  • how key features on the original graph $y=f\left(x\right)$

    • transform to different key features on the graph of $y=\frac{1}{f\left(x\right)}$

• In general

  • the larger the heights of points on $y=f\left(x\right)$

    • the closer to the $x$-axis they become on $y=\frac{1}{f\left(x\right)}$

    • and vice versa

  • If $y=f\left(x\right)$ is positive

    • then $y=\frac{1}{f\left(x\right)}$ is positive

  • If $y=f\left(x\right)$ is negative

    • then $y=\frac{1}{f\left(x\right)}$ is negative

  • If $y=f\left(x\right)$ is increasing

    • then $y=\frac{1}{f\left(x\right)}$ is decreasing

  • If $y=f\left(x\right)$ is decreasing

    • then $y=\frac{1}{f\left(x\right)}$ is increasing

• More specifically

  • If $y=f\left(x\right)$ has a $y$-intercept at $\left(0,c\right)$ where $c\ne 0$

    • $y=\frac{1}{f\left(x\right)}$ has a $y$-intercept at $\left(0,\frac{1}{c}\right)$

  • If $y=f\left(x\right)$ has an $x$-intercept (root)at $\left(a,0\right)$

    • $y=\frac{1}{f\left(x\right)}$ has a vertical asymptote at $x=a$

  • If $y=f\left(x\right)$ has a vertical asymptote at $x=a$

    • $y=\frac{1}{f\left(x\right)}$ has a discontinuity at $\left(a,0\right)$

    • The discontinuity looks like a root when you sketch

  • If $y=f\left(x\right)$ has a local maximum at $\left(x_1,y_1\right)$ where $y_1\ne 0$

    • $y=\frac{1}{f\left(x\right)}$ has a local minimum at $\left(x_1,\frac{1}{y_1}\right)$

  • If $y=f\left(x\right)$ has a local minimum at $\left(x_1,y_1\right)$ where $y_1\ne 0$

    • $y=\frac{1}{f\left(x\right)}$ has a local maximum at $\left(x_1,\frac{1}{y_1}\right)$

  • If $y=f\left(x\right)$ has a horizontal asymptote at $y=k$

    • if $k\ne 0$ then $y=\frac{1}{f\left(x\right)}$ has a horizontal asymptote at $y=\frac{1}{k}$

    • if $k=0$ then $y=\frac{1}{f\left(x\right)}\to \pm \infty$

  • If $y=f\left(x\right)\to \pm \infty$ as $x\to \pm \infty$

    • $y=\frac{1}{f\left(x\right)}$ has a horizontal asymptote at $y=0$