DWG NO. 1.4 — Lesson 4 of 6

Transformations of Functions

Unit 1: Functions and Their Graphs · ~25 min

Objective Graph a function transformed by shifts, stretches, compressions, and reflections, given its parent function.

Rather than plotting points from scratch every time, it's often faster to recognize a function as a parent function — a simple base shape like y = x² or y = |x| — with something done to it. Each type of change moves or reshapes the graph in a predictable way.

Change to f(x)	Effect on the graph
f(x) + k	Vertical shift up k units (k > 0) or down (k < 0)
f(x − h)	Horizontal shift right h units (h > 0) or left (h < 0)
a · f(x)	Vertical stretch (\|a\| > 1) or compression (0 < \|a\| < 1)
f(a · x)	Horizontal compression (\|a\| > 1) or stretch (0 < \|a\| < 1)
−f(x)	Reflection over the x-axis
f(−x)	Reflection over the y-axis

Notice the trap in the second row: h inside the parentheses moves the graph in the opposite direction you'd guess. f(x−2) shifts right, not left, because it takes an x-value 2 larger to produce the same output the parent function gave at the original x.

Order of operations for multiple transformations

When a function combines several changes, apply them in the same order you would evaluate the expression: horizontal changes (stretch/compress, then shift) first, then vertical changes (stretch/compress, then shift) — or just track what happens to a few key points, like the vertex, one at a time.

Parent y = x² (dim) shifted right 2 and up 1 to form g(x) = (x−2)² + 1 (gold)

Worked Example 1 · Shift and vertex tracking

ProblemDescribe how g(x) = (x−2)² + 1 transforms the parent function f(x) = x².

1The "−2" inside the parentheses shifts the graph right 2 units.

2The "+1" outside shifts the graph up 1 unit.

3The parent vertex (0, 0) moves to (2, 1).

g(x) is f(x) shifted right 2 and up 1; new vertex (2, 1)

Worked Example 2 · Stretch and reflection

ProblemDescribe how g(x) = −2|x| transforms the parent function f(x) = |x|.

1The factor of 2 stretches the graph vertically, making it rise twice as fast.

2The negative sign in front reflects the graph over the x-axis, so it opens downward instead of upward.

g(x) is f(x) vertically stretched by 2 and reflected over the x-axis

Worked Example 3 · Combining three transformations

ProblemDescribe g(x) = 3(x+1)² − 4 as a transformation of f(x) = x².

1"(x+1)" shifts the graph left 1 unit.

2The factor of 3 stretches the graph vertically.

3"−4" shifts the graph down 4 units. The vertex (0,0) moves to (−1, −4).

g(x): f(x) shifted left 1, stretched vertically by 3, shifted down 4; vertex (−1, −4)

Guided practice

1.Describe how g(x) = x² − 5 transforms the parent f(x) = x².
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Vertical shift down 5 units. The vertex moves from (0,0) to (0,−5).
2.Describe how g(x) = \(\sqrt{x+4}\) transforms the parent f(x) = \(\sqrt{x}\).
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Horizontal shift left 4 units. Remember: +4 inside the argument moves the graph left, not right.
3.Describe how g(x) = −x³ transforms the parent f(x) = x³.
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Reflection over the x-axis. Every output flips sign; where f rose, g falls.
4.Given f(x) = x² has vertex (0,0), find the vertex of g(x) = 2(x−3)² + 7 without graphing.
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Shift right 3, stretch by 2 (doesn't move the vertex), shift up 7. New vertex: (3, 7)
5.Describe, in order, all the transformations that turn f(x) = |x| into g(x) = ½|x+2| − 1.
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Shift left 2, vertically compress by a factor of ½, shift down 1. The vertex moves from (0,0) to (−2,−1).

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