Function transformations let you predict how a graph changes without making a full table of values. Starting from a parent function y = f(x), you can shift, stretch, compress, or reflect the graph using changes in the equation. This skill matters because it connects algebraic expressions to visual patterns on a coordinate grid.
It also helps you graph complicated functions quickly and accurately.
Key Facts
- Vertical shift: y = f(x) + k moves the graph up k units if k > 0 and down |k| units if k < 0.
- Horizontal shift: y = f(x - h) moves the graph right h units if h > 0 and left |h| units if h < 0.
- Vertical stretch or compression: y = a f(x) multiplies all y-values by a.
- Reflection across the x-axis: y = -f(x) changes every y-value to its opposite.
- Reflection across the y-axis: y = f(-x) changes every x-value to its opposite.
- General transformed form: y = a f(b(x - h)) + k, where a controls vertical scale and reflection, b controls horizontal scale and reflection, h shifts horizontally, and k shifts vertically.
Vocabulary
- Parent function
- A parent function is the simplest basic function in a family, such as y = x^2, y = |x|, or y = sqrt(x).
- Transformation
- A transformation is a change to a graph's position, shape, size, or orientation.
- Vertical shift
- A vertical shift moves a graph up or down by adding a constant outside the function.
- Horizontal shift
- A horizontal shift moves a graph left or right by adding or subtracting inside the function's input.
- Reflection
- A reflection flips a graph across a line such as the x-axis or y-axis.
Common Mistakes to Avoid
- Treating y = f(x - 3) as a shift left 3 units is wrong because changes inside the input work in the opposite direction, so it shifts right 3 units.
- Confusing y = 2f(x) with y = f(2x) is wrong because 2f(x) stretches vertically, while f(2x) compresses horizontally.
- Forgetting that y = -f(x) reflects across the x-axis is wrong because the negative sign outside the function changes the sign of every y-value.
- Applying transformations in a random order can give wrong points because shifts, reflections, and scale changes affect coordinates in specific ways.
Practice Questions
- 1 The parent function is f(x) = x^2. Write the equation for the graph shifted right 4 units and up 2 units, then find the new vertex.
- 2 For f(x) = |x|, graph or describe y = -2f(x + 3) + 1. State the vertex, direction of opening, and vertical stretch factor.
- 3 A graph of y = f(x) is transformed into y = f(-x) + 5. Explain in words what happens to the graph and why the order of inside and outside changes matters.