Transformation Of Graph Dse Exercise -

Scaling changes the "steepness" or width of the graph.

A. Vertical Scaling

  • Mechanism: All $y$-coordinates are multiplied by a factor of $k$. The $x$-intercepts remain the same.

    • B. Horizontal Scaling

  • Mechanism: The effect on the x-axis is inverse to the value of $k$. The $y$-intercepts remain the same.

    • | New equation | Meaning | |--------------|---------| | (y = f(x-a) + b) | Right a, up b | | (y = -f(x)) | Reflect in x-axis | | (y = f(-x)) | Reflect in y-axis | | (y = k f(x)) | Vertical stretch (k>1) / compress (0<k<1) | | (y = f(ax)) | Horizontal compress (a>1) / stretch (0<a<1) | | (y = f(ax + b)) | First factor: (a(x + b/a)) → compress by (1/a), then shift left (b/a) |

    Order: Horizontal stretch/compress → Horizontal shift → Vertical stretch/compress → Reflection → Vertical shift. transformation of graph dse exercise

    Let ( y = f(x) ) be the original function.

    | Transformation | Equation | Effect | |---------------|----------|--------| | Horizontal shift (right (c)) | ( y = f(x - c) ) | Moves graph right by (c) units | | Horizontal shift (left (c)) | ( y = f(x + c) ) | Moves graph left by (c) units | | Vertical shift (up (c)) | ( y = f(x) + c ) | Moves graph up by (c) units | | Vertical shift (down (c)) | ( y = f(x) - c ) | Moves graph down by (c) units | | Reflection in x-axis | ( y = -f(x) ) | Flips vertically | | Reflection in y-axis | ( y = f(-x) ) | Flips horizontally | | Vertical stretch (factor (a>1)) | ( y = a f(x) ) | Stretches vertically | | Vertical compression ((0<a<1)) | ( y = a f(x) ) | Compresses vertically | | Horizontal stretch ((0<a<1)) | ( y = f(ax) ) | Stretches horizontally (careful) | | Horizontal compression ((a>1)) | ( y = f(ax) ) | Compresses horizontally | Scaling changes the "steepness" or width of the graph

    Note for students: Horizontal transformations are "opposite" of intuition: ( f(x+2) ) shifts left, ( f(2x) ) compresses horizontally.

    For ( y = a(x-h)^2 + k ):