Translations and reflections

The idea

• A transformation changes a graph's position, orientation or size without changing its family.
• Every transformation shows up as a change in the equation, and learning which change does what means you never have to plot points.
• The starting graph is called the parent function — $y = x^2$, $y = 2^x$, $y = \frac{1}{x}$ and so on.

Vertical translations

$$y = f(x) + k$$

• Moves the graph UP by $k$ (down if $k$ is negative).
• This one behaves as you expect: $+3$ moves it up 3.
• $y = x^2 + 5$ is the parabola $y = x^2$ shifted up 5, so its vertex moves from $(0,0)$ to $(0, 5)$.

Horizontal translations

$$y = f(x - h)$$

• Moves the graph RIGHT by $h$ (left if $h$ is negative).

• This one behaves the opposite way to how it looks. The minus sign inside gives a shift to the right.

  • $y = (x - 4)^2$ is shifted right 4
  • $y = (x + 4)^2$ is shifted left 4

• Why it is "backwards". To get the same output as before, $x$ must now be larger by $h$ — so every point has moved right.

• The reliable trick: ask what value of $x$ makes the bracket zero. For $(x - 4)$ that is $x = 4$, and that is where the feature has moved to.

Combining both translations

$$y = f(x - h) + k$$

• Moves the graph $h$ right and $k$ up.

• For a parabola this gives the vertex form $y = (x-h)^2 + k$ with vertex at $(h, k)$ — the same rule you already know, now seen as a translation.

• It works identically for every family:

  • $y = \dfrac{1}{x-2} + 3$ is the basic hyperbola moved right 2 and up 3, so its asymptotes move too, to $x = 2$ and $y = 3$.
  • $y = 2^{x-1} + 4$ is the exponential moved right 1 and up 4, so its asymptote becomes $y = 4$.

• Asymptotes translate with the graph. This is the point students most often miss.

Reflection in the $x$-axis

$$y = -f(x)$$

• Flips the graph upside down, over the horizontal axis.
• Every $y$-value changes sign; $x$-values are unchanged.
• $y = -x^2$ is the parabola opening downwards.
• The $x$-intercepts stay put (since $y = 0$ is unchanged by the sign flip), but maxima become minima.

Reflection in the $y$-axis

$$y = f(-x)$$

• Flips the graph left-to-right, over the vertical axis.

• Every $x$-value changes sign; $y$-values are unchanged.

• $y = 2^{-x}$ is the exponential growth curve flipped into a decay curve.

• The minus goes inside the function for a $y$-axis reflection, and outside for an $x$-axis reflection. Inside affects $x$; outside affects $y$.

The general rule

• Changes OUTSIDE the function affect $y$, and do what you expect.
• Changes INSIDE the function affect $x$, and do the opposite of what you expect.

Equation	Effect
$f(x) + k$	up $k$
$f(x) - k$	down $k$
$f(x - h)$	right $h$
$f(x + h)$	left $h$
$-f(x)$	reflect in the $x$-axis
$f(-x)$	reflect in the $y$-axis

Describing a transformation

• To describe how one graph relates to another:
  • Identify the parent function.
  • Compare the equations term by term.
  • State each transformation with its direction and size.
• "The graph of $y = -(x-3)^2 + 1$ is $y = x^2$ reflected in the $x$-axis, then translated 3 right and 1 up."
• Order matters when a reflection is involved, so describe the reflection first if the minus applies to the whole function.