Function Transformations and Parent Functions

Vocabulary flashcards covering the definitions, algebraic rules, and visual behaviors of function transformations, stretches, shrinks, reflections, and parent functions.

Vertical Shift (Upward)

A transformation where adding a positive value to the outside of a function, such as $f(x) + 2$, moves the graph up by that number of units.

Vertical Shift (Downward)

A transformation where subtracting a value from the outside of a function, such as $f(x) - 3$, moves the graph down by that number of units.

Horizontal Shift (Right)

A transformation where a value is subtracted inside the function, such as $f(x - 4)$, shifting the graph to the right by setting the inside equal to zero ($x - 4 = 0 \implies x = 4$).

Horizontal Shift (Left)

A transformation where a value is added inside the function, such as $f(x + 3)$, shifting the graph to the left by setting the inside equal to zero ($x + 3 = 0 \implies x = -3$).

Reflection over the x-axis

A transformation produced by placing a negative sign on the outside of a function, such as $-f(x)$.

Reflection over the y-axis

A transformation produced by placing a negative sign on the inside of a function, such as $f(-x)$.

Reflection over the Origin

A transformation produced by placing negative signs both on the outside and on the inside of a function, such as $-f(-x)$.

Vertical Stretch

A transformation resulting from multiplying the outside of a function by a factor greater than $1$ (such as $2f(x)$), which increases or doubles the $y$-values.

Vertical Shrink

A transformation resulting from multiplying the outside of a function by a fraction (such as $\frac{1}{2}f(x)$), which decreases or cuts the $y$-values in half.

Horizontal Shrink

A transformation caused by placing a factor greater than $1$ inside the function, such as $f(2x)$, which reduces the $x$-values by a factor of $2$ for the same $y$-value.

Horizontal Stretch

A transformation caused by placing a fraction inside the function, such as $f(\frac{1}{2}x)$, which increases the $x$-values by a factor of $2$ for the same $y$-value.

Parent Functions

The core set of basic functions reviewed in the lecture: $y = x$, $y = x^2$, $y = x^3$, $y = \sqrt{x}$, $y = \sqrt[3]{x}$, and $y = |x|$.

Vertex of a Parabola

The central starting point of a parabola determined by its horizontal and vertical shifts, such as the point $(-2, 3)$ for the function $y = -(x + 2)^2 + 3$.

Downward-Opening Parabola

A parabola whose parent function $y = x^2$ has a negative sign in front, such as $y = -(x + 2)^2 + 3$, causing it to reflect over the x-axis and open downward.