⚙️ Function Kingdom · Functions
Function Transformations
Shift, stretch and reflect a parent graph, and read the transformation directly off the equation.
In short
- Every transformed graph has the form y = a f(x - h) + k, with its vertex at (h, k).
- Horizontal shifts run opposite to the sign inside the bracket: (x - 3) moves 3 to the right.
- A number in front stretches vertically, and a negative one also reflects the graph in the x-axis.
- h and k move the graph without changing its shape; a changes the shape without moving the vertex.
Parent functions and their children
A parent function is the simplest member of a family. Two of them matter most at this stage:
- f(x) = x2, whose graph is a parabola with its vertex at (0, 0)
- f(x) = |x|, whose graph is a V with its corner at (0, 0)
Every other member of the family is the parent graph picked up, moved, flipped or stretched. Rather than plotting points all over again, you learn to read those changes straight off the equation:
y = a f(x - h) + k
Three numbers control everything. h moves it sideways, k moves it up or down, and a stretches or flips it vertically. The vertex (or corner) lands at (h, k).
Translations: the sideways surprise
Vertical shifts behave exactly as you would expect. Adding k on the outside lifts the whole graph by k:
y = x2 + 4 is the parabola raised 4 units y = x2 - 4 is the parabola lowered 4 units
Horizontal shifts run the opposite way to the sign inside the bracket:
y = (x - 3)2 moves the graph 3 units to the RIGHT y = (x + 3)2 moves it 3 units to the LEFT
That feels backwards until you ask which x makes the bracket zero. For (x - 3)2 the bracket is zero at x = 3, and zero is where the parent had its vertex — so the vertex has moved to x = 3.
Put the two together: y = (x + 2)2 - 5 is the parabola shifted 2 left and 5 down, with its vertex at (-2, -5).
Reflections and stretches
The number a in front multiplies every output, so it acts vertically.
- a > 1 stretches the graph vertically, making it narrower and steeper. y = 3x2 rises three times as fast as y = x2.
- 0 < a < 1 compresses it, making it flatter.
- a < 0 reflects it in the x-axis: everything that pointed up now points down.
So y = -2x2 is the parabola stretched by 2 and turned upside down. Its vertex is still at (0, 0), but it is now the highest point rather than the lowest.
Note that a changes the shape but never moves the vertex, and h and k move the vertex but never change the shape. That separation is what makes the form y = a f(x - h) + k so convenient.
Reading and writing the equation
From an equation to a description. Take the three numbers in turn.
y = -2(x - 1)2 + 7 a = -2 -> stretched by 2 and reflected in the x-axis h = 1 -> moved 1 to the right k = 7 -> moved 7 up vertex at (1, 7), and since a < 0 it is a maximum
From a description to an equation. Build it in the same order: start with the parent, replace x by (x - h), multiply by a, add k.
"|x| moved 4 left, 3 down, reflected in the x-axis" becomes y = -|x + 4| - 3, with its corner at (-4, -3).
Whenever you are unsure of a sign, test the vertex. Substituting x = h should give y = k. That single check catches the horizontal-shift error almost every time.
Worked examples
Example 1
Describe how the graph of y = (x + 5)2 - 2 relates to y = x2, and give its vertex.
- Inside the bracket: x + 5 is zero when x = -5, so the graph moves 5 units LEFT.
- Outside: the -2 lowers the whole graph by 2 units.
- There is no number in front, so a = 1: no stretch and no reflection.
- The vertex is at (-5, -2).
Example 2
Write the equation of y = x2 after moving 3 right, 4 up, and reflecting in the x-axis.
- A move of 3 to the right replaces x by (x - 3), giving (x - 3)2.
- The reflection in the x-axis puts a minus sign in front: -(x - 3)2.
- Moving 4 up adds 4 at the end: y = -(x - 3)2 + 4.
- Check the vertex: at x = 3 the bracket is 0, so y = 4, and the vertex is (3, 4) as expected.
Practice problems, with solutions
Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.
Problem 1
Difficulty 1 of 5The graph of f(x) = x2 is moved 2 units to the right. Which equation describes the new graph?
- none of these (0)
- y = (x - 2)2
- y = (x + 2)2
Answer: B. y = (x - 2)2
- A move of 2 to the right replaces x by (x - 2).
- There is no vertical shift.
- y = (x - 2)2
Problem 2
Difficulty 3 of 5y = -|x + 8| + 8 What are the coordinates of the corner point of this graph? Give your answer as the pair (x, y).
Answer: (-8, 8)
- The form is y = a(x - h) + k with a = -1, h = -8, k = 8.
- x + 8 = 0 when x = -8.
- Then y = -1 x 0 + 8 = 8.
- Vertex: (-8, 8)
Problem 3
Difficulty 4 of 5The graph of f(x) = |x| becomes the graph of y = |x - 8|. How has it moved?
- 0 units down
- 8 units left
- 0 units up
- 8 units right
Answer: D. 8 units right
- Inside the bracket: x - 8 is zero when x = 8, so the graph slides 8 units right.
- Nothing is added outside, so there is no vertical movement.
- Transformation: 8 units right.
Common mistakes
- Reading (x + 4)2 as a shift 4 to the right instead of 4 to the left.
- Writing the vertex as (k, h) instead of (h, k).
- Dropping or adding a minus sign, so the graph opens the wrong way.
- Thinking a vertical stretch also moves the graph up.
What you should be able to do
- Describe the effect of f(x) + k and f(x + h).
- Describe vertical stretches, compressions and reflections.
- Write the equation of a transformed parent function.
- Sketch a transformed graph from its equation.
Where this fits in the curriculum
Common Core
- HSF-BF.B.3
High school — Identify the effect on a graph of replacing f(x) by f(x) + k, k·f(x), f(kx) and f(x + k).
- HSF-IF.C.7
High school — Graph functions expressed symbolically and show key features of the graph.
SAT
- Passport to Advanced Math
Interpreting and manipulating non-linear functions, including shifts of a graph.
Transformations are tested through the questions rather than named as a topic; this is the domain those questions are counted in.