Integral Calculus, HW 2.1.1 | University of Texas at Austin (OnRamps)
Tags: function transformations, shifts, translations, reflections, vertical stretch, vertical compression, toolkit functions, parent functions, f(x), graph transformations
Difficulty: Introductory to Intermediate | Prerequisites: Familiarity with basic function notation, f(x), and plotting points on the Cartesian plane. Review parent/toolkit functions (linear, quadratic, absolute value, square root, cubic, reciprocal, exponential) before starting.
Function transformations are the toolkit that lets you take a single parent function and generate an entire family of related graphs by shifting, stretching, compressing, or reflecting it. This sits at the heart of precalculus and early calculus: once you see how a base graph moves, you can sketch complicated-looking functions in seconds without plotting dozens of points.
If you missed the earlier lectures on parent functions (the basic shapes of x², √x, |x|, 1/x, 2ˣ, x³), go back and review those first. Every transformation problem assumes you already know what the unmodified graph looks like. This unit builds the bridge between knowing a handful of shapes and being able to graph nearly anything a textbook throws at you.
Every transformation of f(x) comes down to four moves: shift it (left/right/up/down), stretch or compress it (vertically or horizontally), or reflect it (over the x-axis or y-axis). Changes inside the function's argument (inside the parentheses) affect the x-direction and work in the opposite way you might expect. Changes outside the function affect the y-direction and work the way you would expect.
Parent function (toolkit function)
The simplest form of a function family, before any transformation is applied. Examples: f(x) = x², f(x) = |x|, f(x) = √x, f(x) = 1/x, f(x) = 2ˣ.
Think of it as the "default" graph you start from before anything gets moved, stretched, or flipped.
Vertical shift (vertical translation)
Adding or subtracting a constant outside the function: f(x) + k shifts the graph up by k units, f(x) − k shifts it down.
In simple terms, this means the whole graph slides up or down without changing shape.
Horizontal shift (horizontal translation)
Replacing x with (x − h) inside the function: f(x − h) shifts the graph right by h units, f(x + h) shifts it left by h units.
In simple terms, the graph slides left or right. The direction is the opposite of the sign you see, which trips up most students.
Reflection over the x-axis
Multiplying the entire function by −1: −f(x). Every y-value flips sign, so the graph turns upside down.
Think of it as flipping the graph like a pancake over the horizontal axis.
Reflection over the y-axis
Replacing x with −x inside the function: f(−x). The graph becomes its own mirror image across the vertical axis.
Think of it as holding the graph up to a mirror placed along the y-axis.
Vertical stretch
Multiplying the function by a constant a where a > 1: a · f(x). Every y-value gets scaled up, so the graph gets taller.
In simple terms, the graph is pulled away from the x-axis.
Vertical compression
Multiplying the function by a constant a where 0 < a < 1: a · f(x). Every y-value gets scaled down, so the graph gets shorter/flatter.
In simple terms, the graph is pushed toward the x-axis.
Horizontal stretch
Replacing x with (1/b)x inside the function where 0 < b < 1, or equivalently f(bx) with 0 < b < 1. The graph widens.
Think of it as pulling the graph sideways away from the y-axis.
Horizontal compression
Replacing x with bx inside the function where b > 1: f(bx). The graph narrows.
Think of it as squeezing the graph sideways toward the y-axis.
Vertical shifts: f(x) + k
f(x) + k shifts the graph UP by k units (k > 0)
f(x) − k shifts the graph DOWN by k units
The shape of the graph does not change, only its vertical position
Example from HW: f(x) + 10 means every point on the original graph moves 10 units upward
Example from HW: f(x) + 4 means every point moves 4 units upward
Horizontal shifts: f(x − h) and f(x + h)
f(x − h) shifts the graph RIGHT by h units
f(x + h) shifts the graph LEFT by h units
The key insight: the sign inside the parentheses is the opposite of the direction of the shift
Example from HW: f(x − 73) shifts the graph 73 units to the RIGHT
Example from HW: f(x + 39) shifts the graph 39 units to the LEFT, because x + 39 = x − (−39)
Example from HW: f(x + 7) shifts left 7, f(x − 13) shifts right 13
Combined shifts
When writing a formula for a shifted function, apply both shifts together.
f(x) = √x shifted up 4 and left 3 becomes: √(x + 3) + 4
Left 3 means replace x with (x + 3)
Up 4 means add 4 outside
f(x) = |x| shifted down 7 and right 2 becomes: |x − 2| − 7
f(x) = 1/x shifted down 9 and right 1 becomes: 1/(x − 1) − 9
f(x) = 1/x² shifted up 6 and left 10 becomes: 1/(x + 10)² + 6
Reading shifts from tables (Problems 15, 16)
Compare the x-values and y-values of g(x) and h(x) to f(x).
If every x-value in g(x) is 1 more than in f(x) but the outputs are identical, that is a horizontal shift right by 1: g(x) = f(x − 1)
If the outputs of h(x) are each 1 more than f(x) at the same x-values, that is a vertical shift up by 1: h(x) = f(x) + 1
Reflection over the x-axis: −f(x)
Multiply the entire output by −1
Every y-value changes sign: positive becomes negative, negative becomes positive
The graph flips upside down
Example from HW: −f(x) is the original graph reflected over the x-axis
Example: starting with f(x) = 3ˣ, reflecting over the x-axis gives −3ˣ
Reflection over the y-axis: f(−x)
Replace every x with −x inside the function
The graph becomes its mirror image across the y-axis
Example from HW: f(−x) is the original graph reflected over the y-axis
Example: starting with f(x) = 2ˣ, reflecting over the y-axis gives 2⁻ˣ (which equals (1/2)ˣ)
Combining reflections with other transformations
Starting with f(x) = 3ˣ, reflecting over the x-axis AND the y-axis gives −3⁻ˣ
Starting with f(x) = 3ˣ, reflecting over the x-axis, shifting left 6 and down 11 gives −3⁽ˣ⁺⁶⁾ − 11
Starting with f(x) = 5ˣ, reflecting over the x-axis gives −5ˣ
Starting with f(x) = 5ˣ, reflecting over the y-axis, shifting right 2 and up 9 gives 5⁻⁽ˣ⁻²⁾ + 9
Order matters when you combine reflections with shifts. Apply reflections and stretches first, then shifts.
Vertical stretch: a · f(x) where a > 1
Every y-value is multiplied by a
The graph gets taller (pulled away from the x-axis)
Example from HW: 3f(x) − 3 means the graph is vertically stretched by a factor of 3, then shifted down 3
Example from HW: 7f(x) means the graph is vertically stretched by a factor of 7
Example from HW: 2f(x) means the graph is vertically stretched by a factor of 2
Vertical compression: a · f(x) where 0 < a < 1
Every y-value is multiplied by a fraction
The graph gets flatter (pushed toward the x-axis)
Example: (1/5)x³ means the cubic function is vertically compressed by a factor of 5
Example: (1/2)|x − 10| means the absolute value function is vertically compressed by a factor of 2
Combining vertical scaling with a negative sign
−2ˣ + 1 means: take 2ˣ, reflect it over the x-axis, then shift up 1
8f(−x) means: reflect over the y-axis, then vertically stretch by a factor of 8
−f(8x) means: horizontally compress by a factor of 8, then reflect over the x-axis
Master transformation formula
y = a \cdot f\bigl(b(x - h)\bigr) + kWhere:
a = vertical stretch/compression and x-axis reflection (negative a flips over x-axis)
b = horizontal stretch/compression and y-axis reflection (negative b flips over y-axis)
h = horizontal shift (positive h shifts right, negative h shifts left)
k = vertical shift (positive k shifts up, negative k shifts down)
Quick reference for reading transformations from an equation
What you see | What it does |
|---|---|
f(x) + k | Shift up k |
f(x) − k | Shift down k |
f(x − h) | Shift right h |
f(x + h) | Shift left h |
−f(x) | Reflect over x-axis |
f(−x) | Reflect over y-axis |
a · f(x), a > 1 | Vertical stretch by factor a |
a · f(x), 0 < a < 1 | Vertical compression by factor 1/a |
f(bx), b > 1 | Horizontal compression by factor b |
f(bx), 0 < b < 1 | Horizontal stretch by factor 1/b |
Students often think f(x + 3) shifts the graph to the right. It does not. f(x + 3) shifts LEFT by 3. The sign inside the parentheses works opposite to what you might expect. Think of it this way: the function reaches the same y-value 3 units sooner (to the left).
Students often confuse vertical and horizontal transformations. A number multiplied outside, like 3f(x), stretches vertically. A number multiplied inside, like f(3x), compresses horizontally. Outside = vertical. Inside = horizontal.
Students sometimes apply shifts before reflections/stretches when writing equations. The standard order is: apply stretches, compressions, and reflections first, then apply shifts. Getting this wrong changes the final equation.
Students often forget that horizontal transformations are "opposite" in direction AND in stretch/compress behaviour. f(2x) does not stretch horizontally, it compresses. f((1/2)x) does not compress horizontally, it stretches.
⚠️ Describing transformations in words from an equation (Problems 1 to 10, 37 to 46) is a near-certain exam question. Be able to go both directions: equation to words, and words to equation.
⚠️ Writing a formula from a verbal description (Problems 11 to 14, 51 to 56) tests whether you understand the sign conventions for shifts and the order of operations for combined transformations.
⚠️ Reading transformations from tables of values (Problems 15 to 16) tests conceptual understanding rather than formula memorisation. If x-values shift, it is horizontal. If y-values shift, it is vertical.
⚠️ Sketching transformed graphs of exponential functions (Problems 17 to 22) and toolkit functions (Problems 23 to 26) is a classic exam format: you are given the parent graph and asked to draw the transformation.
True or false: f(x − 5) shifts the graph 5 units to the left. (False. It shifts right.)
True or false: −f(x) reflects the graph over the y-axis. (False. It reflects over the x-axis. f(−x) reflects over the y-axis.)
Fill in the blank: 4f(x) is a vertical ______ by a factor of 4. (stretch)
True or false: f(3x) compresses the graph horizontally by a factor of 3. (True.)
Fill in the blank: f(x) + 7 shifts the graph ______ by 7 units. (up)
Q: Describe the transformation: f(x − 73)
A: The graph of f(x) is shifted 73 units to the right.
Q: Describe the transformation: f(x) + 10
A: The graph of f(x) is shifted 10 units up.
Q: Describe the transformation: 3f(x) − 3
A: The graph of f(x) is vertically stretched by a factor of 3, then shifted down 3 units.
Q: Write a formula for f(x) = √x shifted up 4 units and left 3 units.
A: √(x + 3) + 4
Q: Write a formula for f(x) = |x| shifted down 7 units and right 2 units.
A: |x − 2| − 7
Q: Starting with f(x) = 3ˣ, write the equation after reflecting about the x-axis and the y-axis.
A: −3⁻ˣ
Q: Describe the transformation: −f(x)
A: The graph of f(x) is reflected over the x-axis.
Q: Describe the transformation: f(−2x)
A: The graph of f(x) is reflected over the y-axis and horizontally compressed by a factor of 2.
Q: Describe the transformation: 8f(−x)
A: The graph of f(x) is reflected over the y-axis and vertically stretched by a factor of 8.
Q: If f(x) has the values {(0,0), (1,1), (2,−1), (3,3), (4,4)} and g(x) has the values {(1,0), (2,1), (3,−1), (4,3), (5,4)}, write g(x) as a transformation of f(x).
A: g(x) = f(x − 1). Every x-value in g is 1 more than in f, so the graph shifted right by 1.
This material connects directly to graphing exponential, logarithmic, and trigonometric functions later in the course. Every one of those chapters assumes you can identify shifts, reflections, and stretches on sight.
In calculus, understanding transformations helps with curve sketching and with recognising how derivatives and integrals relate to the parent function's graph. The chain rule, for instance, is essentially differentiation applied to a horizontally transformed function.
If you are also studying inverse functions, note that reflecting over the line y = x (the method for graphing an inverse) is not the same as reflecting over the x-axis or y-axis covered here, but the mechanical thinking is similar.
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