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Math · 28 · Advanced Math

Transformation of Graphs

Notes

Key points

  • f(x)+kf(x)+k: up kk. f(x)−kf(x)-k: down kk.
  • f(x−h)f(x-h): right hh. f(x+h)f(x+h): left hh.
  • Outside changes move y as written; inside changes move x the opposite way.
  • −f(x)-f(x): reflect over the x-axis. f(−x)f(-x): reflect over the y-axis.
  • af(x)af(x): vertical stretch by aa; shrink if 0<a<10<a<1.
  • Track one key point (vertex, intercept) through every change.

Formulas

Vertical shifty=f(x)+k: (x,y)→(x, y+k)y=f(x)+k:\ (x,y)\to(x,\ y+k)
Horizontal shifty=f(x−h): (x,y)→(x+h, y)y=f(x-h):\ (x,y)\to(x+h,\ y)
Reflections−f(x): (x,y)→(x,−y)f(−x): (x,y)→(−x,y)-f(x):\ (x,y)\to(x,-y)\qquad f(-x):\ (x,y)\to(-x,y)
Vertical stretchaf(x): (x,y)→(x, ay)af(x):\ (x,y)\to(x,\ ay)
Horizontal stretchf(bx): (x,y)→(xb, y)f(bx):\ (x,y)\to\left(\tfrac{x}{b},\ y\right)

Common traps

  • Moving f(x+3)f(x+3) right. It moves left 3.
  • Mixing −f(x)-f(x) (flip over x-axis) with f(−x)f(-x) (flip over y-axis).
  • Shifting the x-intercepts for a vertical change, or the y-intercept for a horizontal one.

SAT tip

Unsure of a direction? Graph a sample f and the transformed version in Desmos and watch where it moves.

Test yourself

10 official SAT questions on Transformation of Graphs, easiest first.

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