Function Transformations: Shift, Reflect, Stretch — One Equation Tells You Everything

Once you know the parent function, you can sketch any transformation by reading the equation. Master the four moves — vertical/horizontal shift, reflection, stretch — and you'll graph every quadratic, root, and absolute-value function on sight.

9 phút TEKS 2A,2B,2C,2D Algebra 2

Every function family has a "parent"

Algebra 2 introduces six parent functions: linear, quadratic, square root, cubic, cube root, and absolute value. Every more-complicated function is just a parent function with shifts, reflections, or stretches applied. Read those moves out of the equation and you can sketch the graph without plotting a single point.

Big idea

Inside the parentheses (with x): affects horizontal direction — and reverses your intuition. Outside the parentheses (with y): affects vertical direction — and matches your intuition.

The six parent functions

Three of the six parents. Cube root, cubic, and absolute value follow the same logic but with different shapes.

Linear: f(x) = x: Straight line through origin, slope 1.
Quadratic: f(x) = x²: U-shaped parabola opening up, vertex at origin.
Square Root: f(x) = √x: Half-curve starting at origin, defined only for x ≥ 0.
Cubic: f(x) = x³: S-shape through origin; opposite ends go opposite directions.
Cube Root: f(x) = ∛x: Sideways S through origin; defined for all real x.
Absolute Value: f(x) = |x|: V-shape with vertex at origin, opening up.

The four transformations

g(x) = a · f(b(x − h)) + k h shifts horizontally (right if h > 0) k shifts vertically (up if k > 0) a stretches/reflects vertically (negative = flip across x-axis) b stretches/reflects horizontally (negative = flip across y-axis)

Inside the parentheses is reversed

(x − 3) shifts the graph 3 units right, not left. The minus sign is built into the formula. Same for (x + 3) — that means h = −3, so the graph shifts 3 units left.

Worked example: shifts

The graph of f(x) = x² is shifted 3 units right and 4 units down. Find the new equation.

Right by 3 → replace x with (x − 3) Down by 4 → subtract 4 from the whole function g(x) = (x − 3)² − 4

Practice

Apply two shifts

The graph of f(x) = x² is shifted 3 units right and 4 units down. The new equation is:

Open the question →

Reflections

y = −f(x): reflect across x-axis y = f(−x): reflect across y-axis Negative on the OUTPUT flips up/down. Negative on the INPUT flips left/right.

Practice

Identify the right reflection

Which transformation reflects the graph of y = f(x) across the x-axis?

Open the question →

Function composition

(f ∘ g)(x) means "f of g of x" — apply g first, then plug the result into f.

f(x) = x + 3, g(x) = x² (f ∘ g)(2) = f(g(2)) = f(4) = 7 Order matters: (g ∘ f)(2) = g(f(2)) = g(5) = 25, which is different.

Practice

Compose two functions

If f(x) = x + 3 and g(x) = x², what is (f ∘ g)(2)?

Open the question →

3-second recap

Outside the function = vertical move (matches intuition)
Inside the function = horizontal move (reversed)
Negative outside → flip across x-axis. Negative inside → flip across y-axis.
(f ∘ g)(x) = f(g(x)) — work right to left.