Function Transformations

Function Transformations: Key Concepts and Examples

Introduction to Function Transformations

• Transforming functions involves altering their graphs through various operations.
• Transformations include sliding (translations), stretching/compressing, and reflections.
• The core principle: To determine the new function's formula after a transformation, identify what operation is performed on the coordinates (x, y) and then apply the opposite operation to the function's equation.

Translations (Sliding)

Horizontal Translations

• Sliding to the Right: To shift a function's graph $h$ units to the right, replace $x$ with $(x - h)$ in the function's equation.
  • Example: If $y = f(x)$, shifting it 2 units to the right results in $y = f(x - 2)$.
• Sliding to the Left: To shift a function's graph $h$ units to the left, replace $x$ with $(x + h)$ in the function's equation.
  • Example: If $y = f(x)$, shifting it 3 units to the left results in $y = f(x + 3)$.

Vertical Translations

• Sliding Up: To shift a function's graph $k$ units upward, replace $y$ with $(y - k)$ in the function's equation.
  • Example: If $y = f(x)$, shifting it 1 unit up results in $y - 1 = f(x)$, which can be rewritten as $y = f(x) + 1$.
• Sliding Down: To shift a function's graph $k$ units downward, replace $y$ with $(y + k)$ in the function's equation.

Combining Horizontal and Vertical Translations

• To slide a function left by 3 units and up by 1 unit, transform $y = f(x)$ to $y - 1 = f(x + 3)$.

Reflections

Reflection Over the x-axis

• To reflect a function over the x-axis, multiply the y-coordinates by $-1$. In the equation, replace $y$ with $\frac{y}{-1}$ (or $-y$).
  • Example: Given $y = -x + 1$, reflecting over the x-axis involves replacing $y$ with $\frac{y}{-1}$, leading to $\frac{y}{-1} = -x + 1$. Multiplying both sides by $-1$ gives $y = x - 1$.
  • The original function $y = -x + 1$ has a y-intercept of 1 and a slope of $-1$. The reflected function $y = x - 1$ has a y-intercept of $-1$ and a slope of 1.

Reflection Over the y-axis

• To reflect a function over the y-axis, multiply the x-coordinates by $-1$. In the equation, replace $x$ with $\frac{x}{-1}$ (or $-x$).

Stretching/Compressing

Stretching in the y-direction

• To stretch a function in the y-direction by a factor of $a$, multiply the y-coordinates by $a$. In the equation, replace $y$ with $\frac{y}{a}$.
  • Example: Given $y = x^2$, to stretch it so that the point $(1, 1)$ becomes $(1, 2)$, replace $y$ with $\frac{y}{2}$, resulting in $\frac{y}{2} = x^2$. Solving for y gives $y = 2x^2$.

Stretching in the x-direction

• To stretch a function in the x-direction by a factor of $b$, multiply the x-coordinates by $b$. In the equation, replace $x$ with $\frac{x}{b}$.
  • If the original width of a function is 6, and you want to double it to 12, replace $x$ with $\frac{x}{2}$ in the original function's equation.

The Big Idea: Opposite Operations

• Identify the transformation's effect on the coordinates (x, y).
• Apply the opposite operation to the corresponding variable in the function's equation.
• Opposite of addition is subtraction, and vice versa.
• Opposite of multiplication is division, and vice versa.

Summary Table