A transformation of a function involves shifting, stretching, compressing, or reflecting its graph. These modifications alter the original function's appearance but not its basic shape.
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Vertical shifts occur when you add or subtract a constant from the function: $f(x) + c$ or $f(x) - c$.
Horizontal shifts happen when you add or subtract a constant inside the argument: $f(x+c)$ or $f(x-c)$.
Vertical stretching and compressing are achieved by multiplying the function by a constant: $af(x)$ where $a > 1$ stretches and $0 < a < 1$ compresses.
Horizontal stretching and compressing occur by modifying the input variable with a constant factor: $f(bx)$ where $0 < b < 1$ stretches and $b > 1$ compresses.
Reflection across the x-axis is done by negating the function: $-f(x)$, while reflection across the y-axis is performed by negating the input variable: $f(-x)$.
Review Questions
What happens to the graph of a function when you replace $f(x)$ with $f(x - h) + k$?
How do you reflect a function across the y-axis?
What is the effect of multiplying a function by a constant greater than one?
Related terms
Vertical Shift: A transformation that moves the graph of a function up or down without changing its shape.
Horizontal Shift: A transformation that moves the graph of a function left or right without altering its shape.
Reflection: $A transformation that flips the graph of a function over an axis.$