Integral Calculus, HW 2.1.1 | University of Texas at Austin (OnRamps)
Tags: horizontal stretch, horizontal compression, combined transformations, multi-step transformations, order of transformations, toolkit function graphs, writing equations from graphs
Difficulty: Intermediate | Prerequisites: Part 1 of these study notes (shifts, reflections, vertical stretches/compressions). You should be comfortable with vertical transformations before tackling horizontal scaling and multi-step combinations.
Part 1 covered the building blocks: shifting, reflecting, and scaling vertically. This part tackles horizontal scaling (stretches and compressions in the x-direction) and, more importantly, what happens when you stack several transformations together in one equation.
The homework problems in the second half of HW 2.1.1 are where most students start to struggle, because you need to unpack an equation like f(2x − 4) + 6 into its individual steps and get them in the right order. This is also where the course starts asking you to work backwards: given a graph, write the equation.
If you can reliably decompose a multi-step transformation and reconstruct the equation from a graph, you are well prepared for the exam.
Horizontal scaling works opposite to what you expect: f(2x) compresses the graph toward the y-axis, and f((1/2)x) stretches it away. When multiple transformations are combined in one equation, factor and rewrite the inside of the function carefully, then apply in order: horizontal scale and reflect first, then horizontal shift, then vertical scale and reflect, then vertical shift.
Horizontal compression
Replacing x with bx where b > 1 inside the function: f(bx). The graph is squeezed toward the y-axis by a factor of b. Every x-coordinate is divided by b.
In simple terms, f(2x) makes the graph half as wide. f(10x) makes it a tenth as wide.
Horizontal stretch
Replacing x with bx where 0 < b < 1 inside the function: f(bx). The graph is pulled away from the y-axis. Every x-coordinate is divided by b (which makes it larger).
In simple terms, f((1/6)x) makes the graph 6 times as wide. f((1/13)x) makes it 13 times as wide.
Combined (multi-step) transformation
An equation that applies two or more transformations at once, such as f(2x − 4) + 6. To read it correctly, you must separate it into individual operations and apply them in the proper order.
Think of it as a recipe with multiple steps: the order you follow them in determines the final result.
Order of transformations
The standard sequence for applying transformations: (1) horizontal scale and reflect, (2) horizontal shift, (3) vertical scale and reflect, (4) vertical shift. This order comes from how the algebra nests inside the general form y = a · f(b(x − h)) + k.
In simple terms, work from the inside out for horizontal changes, then from the outside for vertical changes.
f(bx) where b > 1: horizontal compression
f(2x) compresses the graph horizontally by a factor of 2 (the graph becomes half as wide)
f(10x) compresses horizontally by a factor of 10
Example from HW: f(2x) − 20 means compress horizontally by 2, then shift down 20
Example from HW: f(−2x) means compress horizontally by 2 AND reflect over the y-axis (the negative sign adds the reflection)
f(bx) where 0 < b < 1: horizontal stretch
f((1/6)x) stretches the graph horizontally by a factor of 6 (the graph becomes 6 times as wide)
f((1/13)x) stretches the graph horizontally by a factor of 13
Example from HW: f((1/3)x + 3) − 8 involves a horizontal stretch by 3, combined with a shift and a vertical shift
Why horizontal transformations are "backwards"
The reason f(2x) compresses rather than stretches is that to produce the same y-value, you now need half the x-value. If f(4) = 7, then for f(2x) to equal 7 you need 2x = 4, so x = 2. The point that was at x = 4 has moved to x = 2: the graph compressed.
Decomposing compound horizontal arguments
When you see something like f(2x − 4), factor out the coefficient of x first:
f(2x − 4) = f(2(x − 2))
This tells you: compress horizontally by 2, then shift right 2
The +6 in f(2x − 4) + 6 is a separate vertical shift: up 6
Similarly, f((1/3)x + 3) = f((1/3)(x + 9)):
Stretch horizontally by 3, then shift left 9
The −8 in f((1/3)x + 3) − 8 is a vertical shift: down 8
Problems 51 to 56 ask you to apply several transformations to a toolkit function. The strategy is always the same: identify the parent function, apply each transformation in the correct order, and write the final equation.
Worked examples from the HW
Problem 51: f(x) = |x| reflected over the y-axis and horizontally compressed by a factor of 3.
Reflect over y-axis: |−x| = |x| (absolute value absorbs the negative)
Horizontally compress by 3: |3x|
Wait: reflecting |x| over the y-axis gives |−x| = |x|, so the reflection has no visible effect. The answer is |3x|.
Problem 52: f(x) = √x reflected over the x-axis and horizontally stretched by a factor of 5.
Reflect over x-axis: −√x
Horizontally stretch by 5: replace x with (1/5)x, giving −√((1/5)x) or equivalently −√(x/5)
Problem 53: f(x) = 1/x² vertically compressed by a factor of 2, shifted left 8 and down 6.
Vertically compress by 2: (1/2) · (1/x²) = 1/(2x²)
Shift left 8: replace x with (x + 8), giving 1/(2(x + 8)²)
Shift down 6: subtract 6, giving 1/(2(x + 8)²) − 6
Problem 54: f(x) = 1/x vertically stretched by a factor of 4, shifted right 1 and up 10.
Vertically stretch by 4: 4/x
Shift right 1: 4/(x − 1)
Shift up 10: 4/(x − 1) + 10
Problem 55: f(x) = x² horizontally compressed by a factor of 7, shifted right 8 and up 5.
Horizontally compress by 7: (7x)²
Shift right 8: (7(x − 8))² = (7x − 56)²
Shift up 5: (7(x − 8))² + 5
Problem 56: f(x) = x² horizontally stretched by a factor of 7, shifted left 2 and down 12.
Horizontally stretch by 7: ((1/7)x)²
Shift left 2: ((1/7)(x + 2))²
Shift down 12: ((1/7)(x + 2))² − 12
Problems 57 to 66: Describing the transformation of a given equation
These problems give you the final equation and ask you to identify the parent function and describe each transformation.
f(x) = 5(x + 16)² − 24: Parent is x². Vertically stretched by 5, shifted left 16, shifted down 24.
g(x) = 7(x + 11)² − 15: Parent is x². Vertically stretched by 7, shifted left 11, shifted down 15.
h(x) = −15|x − 20| − 19: Parent is |x|. Vertically stretched by 15, reflected over x-axis, shifted right 20, shifted down 19.
k(x) = 8√x − 12: Parent is √x. Vertically stretched by 8, shifted down 12.
m(x) = (1/5)x³: Parent is x³. Vertically compressed by a factor of 5.
n(x) = −(1/2)|x − 10|: Parent is |x|. Vertically compressed by 2, reflected over x-axis, shifted right 10.
p(x) = ((1/7)x)² − 14: Parent is x². Horizontally stretched by 7, shifted down 14.
q(x) = ((1/3)x)³ + 12: Parent is x³. Horizontally stretched by 3, shifted up 12.
a(x) = √(−x + 6): Rewrite as √(−(x − 6)). Parent is √x. Reflected over y-axis, shifted right 6.
b(x) = ∛(−x − 21): Rewrite as ∛(−(x + 21)). Parent is ∛x. Reflected over y-axis, shifted left 21.
Problems 27 to 30, 33 to 36, and 67 to 90 ask you to look at a transformed graph and write its equation. This is the reverse skill: instead of going from equation to graph, you go from graph to equation.
Strategy for writing equations from graphs
Identify the parent function. What basic shape does this graph resemble? (parabola = x², V-shape = |x|, S-curve = x³, half-parabola = √x, exponential curve = 2ˣ or 3ˣ, hyperbola = 1/x)
Locate the key point. For a parabola, find the vertex. For absolute value, find the corner. For √x, find the starting point. For 1/x, find where the asymptotes have moved.
Compare the key point's location to where it sits on the parent function. The difference tells you the shifts (h and k).
Check for reflections. Is the graph flipped upside down (x-axis reflection)? Is it flipped left-right (y-axis reflection)?
Check for stretches or compressions. Pick a second recognisable point and see whether the y-values are scaled compared to the parent function.
Working with the given f(x) graph (Problems 67 to 90)
These problems give you a specific f(x) graph and ask you to express each new graph as a transformation of f(x). Look for:
Has the graph shifted left, right, up, or down? Compare specific points.
Has it been flipped (reflected)? If the graph is upside down relative to f(x), that is −f(x).
Has it been stretched or compressed? If the peaks are taller, it has been vertically stretched.
Is it a combination? Write each piece: for example, f(x − 2) + 1 means shifted right 2 and up 1.
Toolkit function graph recognition
Problems 79 to 90 show transformed toolkit functions. The key is recognising which parent function produced the shape:
A U-shaped curve that opens up or down = x² (quadratic)
A V-shaped graph = |x| (absolute value)
An S-shaped curve passing through the origin = x³ (cubic)
A curve starting at a point and increasing with decreasing slope = √x (square root)
A curve that approaches but never touches the axes = 1/x (reciprocal)
A curve that rises or falls steeply and approaches a horizontal asymptote = exponential (2ˣ, 3ˣ)
The general transformation formula
y = a \cdot f\bigl(b(x - h)\bigr) + kStep-by-step order for applying transformations
Start with the parent function f(x)
Horizontal scale/reflect: replace x with bx (b > 1 compresses, 0 < b < 1 stretches, negative b reflects over y-axis)
Horizontal shift: replace x with (x − h) (positive h shifts right, negative h shifts left)
Vertical scale/reflect: multiply by a (|a| > 1 stretches, 0 < |a| < 1 compresses, negative a reflects over x-axis)
Vertical shift: add k (positive k shifts up, negative k shifts down)
Step-by-step order for reading transformations from an equation
Identify the vertical shift (the constant added/subtracted at the end)
Identify the vertical scale/reflect (the coefficient multiplying the whole function)
Factor the inside of the function to the form b(x − h)
Read off the horizontal shift h and horizontal scale b
Factoring the inside: critical step
f(2x − 4) is NOT a shift of 4. Factor first: f(2(x − 2)). The shift is 2 to the right.
f((1/3)x + 3) is NOT a shift of 3. Factor first: f((1/3)(x + 9)). The shift is 9 to the left.
Transformations are how engineers and scientists adapt a single model to different situations. A base equation for heat transfer, signal processing, or population growth gets shifted and scaled to match particular conditions, rather than deriving a new equation from scratch each time.
In computer graphics, every object on screen is positioned, scaled, and rotated using exactly these transformation principles applied to coordinate systems. When a game character moves across the screen, that is a translation; when the camera zooms, that is a scaling transformation.
Students often read f(2x − 4) as a horizontal shift of 4. It is not. You must factor first: f(2(x − 2)), which is a horizontal compression by 2 and a shift right by 2. Failing to factor is the single most common error on combined transformation problems.
Students often confuse horizontal compression with horizontal stretch. f(2x) compresses (narrows); f((1/2)x) stretches (widens). The intuition is backwards from what you might guess, and it is worth memorising until it becomes second nature.
Students sometimes apply transformations in the wrong order when building an equation from a description. The standard order is: scale/reflect first, then shift. If you shift first and then scale, you get a different equation.
Students occasionally forget that reflecting |x| over the y-axis has no visible effect, since |−x| = |x|. Similarly, reflecting x² over the y-axis gives (−x)² = x², which is the same graph. Not every reflection changes the picture.
⚠️ Writing formulas from verbal descriptions involving multiple transformations (Problems 51 to 56) is a high-probability exam question. Practise both horizontal and vertical transformations in combination.
⚠️ Decomposing compound arguments like f(2x − 4) by factoring is a skill tested repeatedly. If you cannot factor the inside reliably, you will misidentify the horizontal shift on every such problem.
⚠️ Describing transformations of given equations (Problems 57 to 66) requires you to identify the parent function and list every transformation in order. Exams often give partial credit for each transformation identified, so knowing the full list matters.
⚠️ Writing equations from graphs (Problems 27 to 30, 67 to 90) tests your ability to work backwards. Practise identifying the parent function and reading off shifts from the graph's key points.
True or false: f(2x − 6) represents a horizontal shift of 6 units to the right. (False. Factor first: f(2(x − 3)), so it is a horizontal compression by 2 and a shift right by 3.)
Fill in the blank: f((1/5)x) stretches the graph horizontally by a factor of ______. (5)
True or false: reflecting f(x) = x² over the y-axis changes the graph. (False. (−x)² = x², so the graph is unchanged.)
Fill in the blank: in the general form y = a · f(b(x − h)) + k, the horizontal shift is determined by ______. (h)
True or false: when writing an equation from a verbal description, you should apply shifts before stretches. (False. Apply stretches/compressions/reflections first, then shifts.)
Q: Describe the transformation f(10x).
A: The graph of f(x) is horizontally compressed by a factor of 10.
Q: Describe the transformation f((1/6)x).
A: The graph of f(x) is horizontally stretched by a factor of 6.
Q: Write a formula for f(x) = 1/x² vertically compressed by a factor of 2, shifted left 8 and down 6.
A: 1/(2(x + 8)²) − 6
Q: Write a formula for f(x) = x² horizontally compressed by a factor of 7, shifted right 8 and up 5.
A: (7(x − 8))² + 5
Q: Decompose f(2x − 4) + 6 into individual transformations.
A: Factor inside: f(2(x − 2)) + 6. Horizontal compression by 2, shift right 2, shift up 6.
Q: What is the parent function of h(x) = −15|x − 20| − 19, and what transformations have been applied?
A: Parent is |x|. Vertically stretched by 15, reflected over x-axis, shifted right 20, shifted down 19.
Q: What is the parent function of p(x) = ((1/7)x)² − 14, and what transformations have been applied?
A: Parent is x². Horizontally stretched by a factor of 7, shifted down 14.
Q: Rewrite a(x) = √(−x + 6) in factored form and describe the transformations.
A: a(x) = √(−(x − 6)). Parent is √x. Reflected over y-axis, shifted right 6.
Q: If you need to horizontally stretch f(x) = x² by a factor of 7 and then shift left 2, what is the equation?
A: ((1/7)(x + 2))². Apply the stretch first (replace x with (1/7)x), then the shift (replace x with (x + 2)).
Q: Why does f(2x) compress the graph rather than stretch it?
A: Because to get the same output, you need half the input. If f(4) = 7, then f(2x) = 7 requires x = 2. The point moved from x = 4 to x = 2, closer to the y-axis.
The factoring technique used to decompose f(2x − 4) into f(2(x − 2)) is the same algebraic skill you use when completing the square in quadratic equations. If you are also studying vertex form of parabolas, that is exactly this: identifying h and k from the equation.
Horizontal scaling appears again in trigonometry, where it determines the period of sine and cosine functions. The b in y = sin(bx) compresses the period by a factor of b, which is the same principle as f(bx) compressing the graph.
In integral calculus, u-substitution is essentially undoing a horizontal transformation. When you substitute u = 2x − 4, you are reversing the compression and shift to get back to the parent function.
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