Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 1 function transformations? Then this Algebra 1 lesson on Shifting, Stretching and Piecewise Functions explains each idea step by step, with worked examples.
Quick answer: Transformations change a parent graph: f(x) + k shifts it vertically, f(x − h) shifts it horizontally, and a · f(x) stretches or flips it. In y = a|x − h| + k the vertex is (h, k). Piecewise functions use a different rule on each interval, while step functions use flat pieces.
Standards: F-BF.B.3, F-IF.C.7b · Level: Grade 9 Algebra 1
How do you move a graph up, down, left or right?
Start with a parent function such as f(x) = x², f(x) = |x| or f(x) = x. Changing the rule transforms the graph without changing its basic shape.
- f(x) + k: moves the graph up k units (down if k is negative).
- f(x − h): moves the graph right h units (left if h is negative).
- a · f(x): stretches vertically by |a|; a negative a also flips the graph over the x-axis.
The change inside the parentheses works backward. (x − 2)² moves right 2, and (x + 2)² moves left 2.
What does an absolute value graph look like?
The absolute value |x| is the distance from x to 0, so it is never negative. Its graph is a V with the vertex at the origin. In y = a|x − h| + k the vertex is (h, k), and the V opens down when a < 0.
How do piecewise and step functions work?
A piecewise function uses different rules on different parts of the domain. To evaluate at a number, first decide which interval contains it, then use only that rule. Open circles mark endpoints that are not included, and closed circles mark endpoints that are included.
A piecewise function whose pieces are horizontal segments, so the graph looks like stairs. Parking fees and shipping prices often work this way: any part of an hour or a pound is charged as a whole one.
How do you graph a transformation step by step?
- Name the parent function and plot its key points.
- Apply the stretch or flip (multiply the y-values by a).
- Apply the horizontal shift, then the vertical shift.
- Check one point by substituting its x-value in the new rule.
g(x) = −2|x + 1| + 3 has vertex (−1, 3) and opens down. At x = 1: −2(2) + 3 = −1, so (1, −1) is on the graph.
Track the vertex first. It tells you the shift, and the sign of a tells you whether the graph opens up or down.
- f(x) + k moves the graph up k units; f(x − h) moves it right h units.
- a · f(x) stretches vertically by |a| and flips over the x-axis if a is negative.
- In y = a|x − h| + k the vertex is (h, k).
- For a piecewise function, find the interval first, then use its rule.
- A step function has horizontal pieces, with open and closed endpoints.
Shifting, Stretching and Piecewise Functions: frequently asked questions
Why does f(x − 3) move the graph right?
The output f(x − 3) equals f(0) when x = 3. Every point of the original graph now appears 3 units further to the right, so the shift is opposite to the sign in the parentheses.
What does a negative a do in a · f(x)?
It multiplies every output by a negative number, which flips the graph over the x-axis. A parabola that opened up now opens down, and its vertex becomes a maximum.
How do you find the vertex of y = a|x − h| + k?
The vertex is the point (h, k). The expression inside the absolute value is zero when x = h, so that is where the V turns, at height k.
How do you evaluate a piecewise function?
Check which interval the input belongs to, then substitute into the rule for that interval only. At a boundary, use the rule whose inequality includes the endpoint.
What do open and closed circles mean on a graph?
A closed circle means the point is part of the graph. An open circle means the graph approaches that point, but the point itself is not included.