Once you know the shape of a few parent graphs, transformations let you sketch hundreds more without plotting points. Every change to the formula moves, flips or stretches the graph in a predictable way. This guide uses f(x) for any function. If the notation is new, read functions explained first.
The parent functions
These are the base shapes everything else is built from.
| Parent | Shape | Key point |
|---|---|---|
| y = x | straight line | (0, 0) |
| y = x² | U-shaped parabola | vertex (0, 0) |
| y = x³ | S-shaped curve | (0, 0) |
| y = √x | half of a sideways parabola | starts at (0, 0) |
| y = ∣x∣ | V shape | vertex (0, 0) |
| y = 1/x | two curves, asymptotes on the axes | none |
| y = 2x | exponential curve | (0, 1) |
Vertical shifts: f(x) + k
Adding k outside the function moves the graph up k units. Subtracting moves it down.
- y = x² + 4 is the parabola moved up 4. Vertex (0, 4).
- y = |x| − 3 is the V moved down 3. Vertex (0, −3).
Horizontal shifts: f(x − h)
Changing the input moves the graph sideways, and the direction is the opposite of what the sign suggests. f(x − h) moves the graph right h units. f(x + h) moves it left.
Why it looks backwards
In y = (x − 3)², the vertex is where the inside equals 0, so x − 3 = 0, which is x = 3. To get the same output as the parent, x has to be 3 bigger. The whole graph sits 3 units to the right.
This is why vertex form, y = a(x − h)² + k, puts the vertex at (h, k). See solving quadratic equations for how to get a quadratic into that form.
Reflections
- −f(x) flips the graph over the x-axis. Every y-value changes sign. y = −x² opens downward.
- f(−x) flips the graph over the y-axis. Every x-value changes sign. y = √(−x) points left instead of right.
Some graphs don’t change under one of these flips. x² and |x| are their own mirror images across the y-axis (they are called even functions).
Stretches and compressions
- a × f(x) stretches the graph vertically by a factor of a. If a > 1 it gets taller and narrower. If 0 < a < 1 it gets flatter and wider.
- f(bx) compresses the graph horizontally by a factor of b. Like horizontal shifts, this one also works backwards: f(2x) squeezes the graph to half its width.
Example Describe y = 3|x|
The V is stretched vertically by 3. From the vertex, it rises 3 for every 1 across instead of 1 for 1.
Example Describe y = ½x²
The parabola is compressed vertically by a factor of ½. It is wider and flatter: at x = 2 it reaches 2 instead of 4.
Summary table
| Change to the formula | Effect on the graph |
|---|---|
| f(x) + k | up k (down if k is negative) |
| f(x − h) | right h (left if written f(x + h)) |
| −f(x) | reflect over the x-axis |
| f(−x) | reflect over the y-axis |
| a·f(x), a > 1 | vertical stretch by a |
| a·f(x), 0 < a < 1 | vertical compression by a |
| f(bx), b > 1 | horizontal compression by 1/b |
Changes outside the function affect y and behave as you’d expect. Changes inside affect x and behave backwards.
Combining transformations
When several changes happen at once, apply them in this order: horizontal shift, then stretch or reflection, then vertical shift. Tracking one key point through each step is the easiest way to stay on course.
Example Describe and sketch y = −2(x + 1)² + 5
- Start with y = x², vertex (0, 0).
- x + 1 inside: shift left 1. Vertex (−1, 0).
- Multiply by 2: vertical stretch, so the parabola is narrower.
- The negative sign: reflect over the x-axis, so it opens downward.
- + 5 outside: shift up 5. Vertex (−1, 5).
- The graph is a narrow, downward parabola with its highest point at (−1, 5).
Example Write the equation: y = √x is shifted 4 left, reflected over the x-axis and shifted down 2
- 4 left: √(x + 4).
- Reflect over the x-axis: −√(x + 4).
- Down 2: y = −√(x + 4) − 2.
- The graph starts at (−4, −2) and heads down to the right.
Using transformations on tests
The SAT and ACT often show a graph of f and ask which graph is f(x − 2) or which equation matches a shifted graph. Find one clear point on the original, such as a vertex or an intercept, apply the change to that point, and pick the answer that passes through the new point. In Desmos you can also type the original and the candidate and compare them directly.
Try a few
Describe how y = (x − 5)² − 1 relates to y = x².
Show answer
Shifted 5 right and 1 down. The vertex is (5, −1).
Where is the vertex of y = |x + 2| + 3?
Show answer
(−2, 3). The inside x + 2 shifts left 2, and + 3 shifts up 3.
Write the equation of y = x³ shifted 2 right and reflected over the x-axis.
Show answer
y = −(x − 2)³.
f(x) passes through (4, 1). What point must y = f(x − 3) + 2 pass through?
Show answer
(7, 3). Move the point 3 right and 2 up.
How is the graph of y = 4 × 2x different from y = 2x?
Show answer
It is stretched vertically by 4, so the y-intercept moves from (0, 1) to (0, 4). It is also the same as 2x + 2, a shift 2 left.
Describe y = −½√(x − 1).
Show answer
Start √x, shift 1 right, compress vertically by ½ and reflect over the x-axis. It starts at (1, 0) and heads down to the right slowly.