Apply vertical and horizontal stretches to functions
Select Base Function and Stretch Type
f(x)=x+, vertical stretch by
f(x)=x+, horizontal stretch by
f(x)=x^2, vertical stretch by
f(x)=x^2, horizontal stretch by
Result
Derivation
Stretch Rules
Vertical stretch by a: g(x) = a * f(x)
Horizontal stretch by a: g(x) = f(x/a)
a > 1: stretch, 0 < a < 1: compression
Point (x,y) -> (x, a*y) vertical, (a*x, y) horiz
A stretch multiplies the output (vertical) or input (horizontal) by a factor. Stretching makes the graph taller or wider. Compression (factor between 0 and 1) squashes it. Key points transform predictably.
⚠Stretch factor must be positive. Factor > 1 stretches, 0 < factor < 1 compresses. Negative factors also reflect.
What Is a Function Stretch?
Stretching a function multiplies the output (vertical stretch) or the input (horizontal stretch) by a constant factor. This changes the graph proportions while preserving the general shape. Vertical stretches affect the range, horizontal stretches affect the domain.
Vertical Stretch
g(x)=a*f(x). Multiply output by a. y-coordinates scaled. x-intercepts remain fixed.
Horizontal Stretch
g(x)=f(x/a). Divide input by a. x-coordinates scaled. y-intercepts remain fixed.
Stretch vs Compression
a>1: stretch (enlarges). 0<a<1: compression (shrinks). They are reciprocal transformations.
Combined Transformations
Stretches can be combined with shifts and reflections. Order matters: stretches first, then shifts.
Teaching Example: f(x)=2x+1, vertical stretch by 3. g(x)=3*(2x+1)=6x+3. Point (1,3) on f -> (1,9) on g. Slope tripled from 2 to 6.
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