A function’s domain is every input it’s allowed to take; its range is every output it can produce. Reading both off a graph is mostly a matter of asking two questions: how far left-right does the graph extend (domain), and how far up-down does it extend (range)?
Transformations — one parent function, many disguises
Most graphs you’ll meet are a familiar “parent” shape (a parabola, a line, a curve) that’s been shifted, flipped, or stretched. f(x) + k shifts the graph up by k. f(x − h) shifts it right by h. −f(x) flips it upside down. Learning these moves once means you can predict dozens of graphs without plotting a single new point — you’re just moving a shape you already know.
Let me show you one
Start with the parent function f(x) = x2, a parabola with its vertex at the origin. What does g(x) = (x − 3)2+ 2 look like? Read the transformation directly off the notation: “x − 3” shifts right 3, and “+ 2” shifts up 2. The vertex moves from (0, 0) to (3, 2) — same parabola shape, just relocated.
Even, odd, or neither — a useful shortcut
A function is even if its graph is a mirror image across the y-axis (like x2); algebraically, f(−x) = f(x). It’s oddif it has 180° rotational symmetry through the origin (like x3); algebraically, f(−x) = −f(x). Knowing which one you have lets you sketch half the graph and get the rest for free.
The mistake I see most
Reading f(x − 3) as a shift leftinstead of right — the minus sign feels like it should mean “subtract from x,” but inside the parentheses it actually shifts the graph the opposite direction from what your instinct says. The fix: test it with one point. Plug in x = 3 and you get f(0), the same value the parent function has at its own starting point — confirming the shift moved everything right.
Want to work through it together?
If reading a function’s behavior straight off its equation still feels like guesswork, that clears up fast once you practice the handful of transformation rules on shapes you already recognize.
