Transformations

With the graph for $y = f(x)$ already known.

Translating/shifting:

  • $y = f(x - c)$
    Replacing $x$ with $x - c$ will translate/shift the graph along the $x$-axis, moving it rightward by $c$.
  • $y = f(x) + c$
    Replacing $y$ with $y - c$ will translate/shift the graph along the $y$-axis, moving it upward by $c$.

Stretching/shrinking:

  • $y = f(a\cdot x)$
    Replacing $x$ with $a\cdot x$ will shrink the graph on the $x$-axis by a factor of $a$.
  • $y = \frac{f(x)}{a}$
    Replacing $y$ with $a\cdot y$ will shrink the graph on the $y$-axis by a factor of $a$.
  • $y = f(\frac{x}{a})$
    Replacing $x$ with $\frac{x}{a}$ will stretch the graph on the $x$-axis by a factor of $a$.
  • $y = a\cdot f(x)$
    Replacing $y$ with $\frac{y}{a}$ will stretch the graph on the $y$-axis by a factor of $a$.

Flipping/mirroring:

  • $y = f(-x)$
    Replacing $x$ with $-x$ will create flipped mirror image with $y = 0$ as the symmetry axis.
  • $y = -f(x)$
    Replacing $y$ with $-y$ will create flipped mirror image with $x = 0$ as the symmetry axis.
  • $y = f^{-1}(x)$ or $x = f(y)$
    Swapping $x$ and $y$ is sometimes no longer a function, but will create the inverse graph which is a flipped mirror image with $x = y$ as the symmetry axis.

Strategy

Strategy for drawing/graphing arbitrary functions (some steps only apply to polinomials):

  1. Find intersections with $x$-axis and $y$-axis
    • Find $f(0)$ and $f(x) = 0$
    • Use Newton’s method to find approximate roots
  2. Find critical points
    • Where $f’(x)$ equals 0 or doesn’t exist
  3. Find local minima and maxima and where function rises or falls
    • Use the sign of $f’(x)$
    • Use the sign of $f^{\prime\prime}(x)$ where $f’(x) = 0$ ($f^{\prime\prime}(x) = 0$ won’t help)
  4. Find inflection points / concavity
    • Where $f^{\prime\prime}(x) = 0$ and sign changes
    • If $f^{\prime\prime}(x) > 0$, function is concave upwards
    • If $f^{\prime\prime}(x) < 0$, function is concave downwards
  5. Find asymptotes
    • Vertical asymptote
      • Look for where function might divide by zero
    • Horizontal asymptote
      • Not applicable if function is only defined on a limited interval
      • Possible if function contains division
      • Attempt limit of $f(x)$ as $x$ approaches $- \infty$ and $\infty$ (left and right)
      • Divide fraction by $x^n$ (above and below) with $n$ the highest power in the numerator
    • Diagonal asymptote
      • Not applicable if function is only defined on a limited interval
      • Same as for horizontal asymptote
      • Use long division te remove division
      • Put formula in the form $f(x) - (mx + c)$