Transformations of graphs
Many graphs are shifted, stretched, or reflected versions of a few basic shapes. Knowing the transformation rules lets you sketch complicated functions quickly.
Parent functions
Basic shapes include the line \(y = x\), the parabola \(y = x^2\), the cubic \(y = x^3\), the square root \(y = \sqrt{x}\), the absolute value \(y = |x|\), and the reciprocal \(y = \frac{1}{x}\). Each has a recognisable graph that transformations move and reshape.
Vertical shifts
Adding a constant outside the function moves the graph up or down: \(y = f(x) + k\) shifts it up k units (down if k is negative). The graph of \(y = x^2 + 3\) is the standard parabola raised 3 units, with vertex (0, 3).
Horizontal shifts
Adding inside the function moves the graph sideways, in the opposite direction to the sign: \(y = f(x - h)\) shifts right h units. So \(y = (x - 2)^2\) has its vertex at (2, 0), and \(y = |x + 5|\) has its corner at (-5, 0). The sign surprise is the most common mistake.
Reflections
A negative sign outside, \(y = -f(x)\), reflects the graph across the x-axis, so \(y = -x^2\) opens downward. A negative inside, \(y = f(-x)\), reflects across the y-axis: \(y = \sqrt{-x}\) points left instead of right.
Stretches and compressions
Multiplying outside by a, \(y = af(x)\), stretches the graph vertically if \(|a| > 1\) and compresses it if \(0 < |a| < 1\). Multiplying inside, \(y = f(bx)\), compresses horizontally by a factor of b when \(b > 1\). So \(y = 3x^2\) is narrower than \(y = x^2\), and \(y = \tfrac{1}{2}x^2\) is wider.
Combining transformations
For \(y = -2(x + 1)^2 + 4\), start with \(y = x^2\): shift left 1, stretch vertically by 2, reflect across the x-axis, then shift up 4. The result opens downward with vertex (-1, 4). Apply horizontal changes, then stretches and reflections, then vertical shifts.
Even and odd functions
A function is even if \(f(-x) = f(x)\), making its graph symmetric about the y-axis, like \(x^2\) and \(|x|\). It is odd if \(f(-x) = -f(x)\), symmetric about the origin, like \(x^3\) and \(\frac{1}{x}\). Most functions, such as \(x^2 + x\), are neither.
Worked example
Describe \(g(x) = \sqrt{x - 4} - 3\). Starting from \(y = \sqrt{x}\), shift right 4 and down 3. The starting point moves from (0, 0) to (4, -3), so the domain is \(x \ge 4\) and the range \(y \ge -3\). The point (9, 3) on \(\sqrt{x}\) moves to (13, 0), an x-intercept of g.
Key takeaways
Learn the parent graphs: lines, x squared, x cubed, square root, absolute value, and 1/x. f(x) + k shifts up k; f(x - h) shifts right h, opposite to the sign. -f(x) reflects across the x-axis and f(-x) across the y-axis. a f(x) stretches vertically when |a| > 1 and compresses when |a| < 1. Apply horizontal changes, then stretches and reflections, then vertical shifts. Even functions are symmetric about the y-axis and odd functions about the origin.
Flashcard
Basic graph before transformations
Check your understanding
How does the graph of \(y = (x - 2)^2\) compare with \(y = x^2\)?