• Think of points first then function (equation)
• Watch out when the compression is conducted after the translation

Linear Equation forms

• standard form: ax+by=c
• slope-intercept form: y=mx+b
• point-slope form: $y-y_1=m(x-x_1)$
• Difficulty:
  • based on points/equation changed, figure out the new equation/points
  • when using general (a, b) to represent the original point
  • new point finding or new equation writing for graph inversed and then stretched

Test

• (4) Graphing: given y=f(x), draw transformed graph
  • translation, reflection, compression/expansion, square root, reciprocal, absolute value
• (2) Given original point, find new point (x, y)
• (1) Transformation terminology
• Translation (shifts) is always at the last
  • Vertical translation ? units up or down
  • Horizontal translation ? units to the left or right
  • reflection (flip) over/about y-axis, x-axis, y=x (inverse)
  • stretches (dilatation)
  • vertical or horizontal compression by a factor of <1
  • vertical or horizontal expansion by a factor of >1
• (4) Graphing the square root, reciprocal or absolute value of a transformed function (2/3)
• (1) Write the equation of a transformed function given a graph (final step will suffice)
• (1) Consider domain & range of a transformed equation

Graphing

When graphing, draw the arrows in both ends

• When drawing and label the vertical and horizontal asymptotes (p not pronounced), extend the ends using the dotted lines
• Congruent: translation & reflection
• Square root transformation
• Combined transformation

When graphing $y=a\sqrt{b(x-h)}+k$

• Make sure you factor before the graphing
• and Y is isolated
• Transformation equation: $y=af(b(x-h))+k$
  • h>0: right; h<0: left
    • point (x, y) -> point $(\frac{x}{b}+h,ay+k)$
  • new point finding would require the above step
  • Square root transformation $y=\sqrt{f(x)}$ graphing
    • Sketch $y=f(x)$ the part on or above the x-axis
    • Invariant point at y=1, 0
  • Pick additional points on original curve as needed and take square root; Connect

Reciprocal transformation

• Graphing $y=\frac{1}{f(x)}$.
• Vertical asymptotes: find the x-int of $y=f(x)$ and draw vertical asymptotes at the points
• Horizontal asymptotes $y=0$
  • Invariant point at y=1, -1
  • Pick additional points on original curve as needed and take reciprocal; Connect
  • Vertical translation will lead to the shift of the horizontal asymptote
  • Absolute value transformation
• Origin - the common point of x-axis and y-axis