SAT function notation and transformations: what f(x − 2) + 1 really means
f(3) is an output, f(a) = 7 is a clue, and f(x − 2) + 1 is a shifted graph. Learn to read function notation fluently and predict exactly how each change moves a graph.
SPSAT Practice Editorial7 min read
The short version
What you'll take away
- f(3) means “the output when the input is 3”; f(a) = 7 means the output is 7 for input a.
- f(x) + k shifts a graph up k; f(x − h) shifts it right h, the opposite of the sign inside.
- To find a shifted point, apply the same shift to its coordinates.
In this article
Read the notation out loud
Function notation names an input and its output. If \(\displaystyle f(x) = 2x - 5\), then \(\displaystyle f(4)\) means “substitute 4 for x”: \(\displaystyle f(4) = 2(4) - 5 = 3\). The parentheses do not mean multiplication.
Reverse questions give the output instead. If \(\displaystyle f(a) = 7\), then \(\displaystyle 2a - 5 = 7\), so \(\displaystyle a = 6\). Read \(\displaystyle f(a) = 7\) as “the input a produces the output 7.”
Shifts: outside moves up, inside moves sideways
Changes outside the function act on outputs, so they move the graph vertically. Changes inside the parentheses act on inputs, so they move the graph horizontally, in the opposite direction of the sign. The graph below shows \(\displaystyle y = x^2\) and \(\displaystyle y = (x - 2)^2 + 1\), which is the same parabola moved right 2 and up 1.
The study lab
f(x) = x² and f(x − 2) + 1
The vertex moves from (0, 0) to (2, 1).
- y = x²
- y = (x − 2)² + 1
The transformation cheat sheet
These five changes cover most SAT transformation questions.
See the skill
What each change does to y = f(x)
f(x) + k
Up k
Each point (x, y) becomes (x, y + k).
f(x − h)
Right h
Each point (x, y) becomes (x + h, y). f(x + 3) moves left 3.
−f(x)
Reflect over the x-axis
Each point (x, y) becomes (x, −y).
a · f(x)
Vertical stretch by a
Each point (x, y) becomes (x, ay).
Your turn
Quick practice
If f(x) = 2x − 5, what is the value of f(4) − f(1)?
Hint: Evaluate each output separately, then subtract.
Practice: try these before checking the answers
The point (2, 7) is on the graph of y = f(x) for questions 4 and 5.
- 1. If \(\displaystyle f(x) = x^2 - 3\), find \(\displaystyle f(-2)\).
- 2. If \(\displaystyle f(x) = 3x + 1\) and \(\displaystyle f(a) = 16\), find a.
- 3. Write the function for the graph of f shifted 4 units left and 2 units down.
- 4. If \(\displaystyle g(x) = f(x) - 3\), what point on g corresponds to (2, 7)?
- 5. If \(\displaystyle h(x) = f(x - 1)\), what point on h corresponds to (2, 7)?
Sources & further reading
One step from here
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