Function Transformations: Shifting Stretching Reflecting

f(x) = x². You know this graph: a parabola opening upward, vertex at the origin.

Now look at g(x) = (x - 2)² + 3.

You could plot points. Or you could recognize: this is the same parabola, shifted right 2 units and up 3 units.

The equation tells you the transformation.

You don't need to plot. You can read the shape from the formula. That's the power of understanding transformations.

The Unlock: Functions Are Objects You Can Move

You're used to thinking of functions as equations you evaluate. Plug in x, get out f(x).

But functions are also shapes in the coordinate plane. And shapes can be moved, stretched, squashed, flipped.

Transformations modify the graph in predictable ways.

Adding a constant shifts the graph vertically. Subtracting from x shifts it horizontally. Multiplying stretches or compresses. Negating flips it.

The algebra encodes the geometry. Once you see the pattern, you can read the transformation directly from the equation.

Four Basic Transformations

There are four fundamental transformations:

1. Vertical shift: f(x) + c
2. Horizontal shift: f(x - c)
3. Vertical stretch/compression: c·f(x)
4. Horizontal stretch/compression: f(cx)

And two more that combine with these:

5. Vertical reflection: -f(x)
6. Horizontal reflection: f(-x)

Master these six, and you can analyze any transformed function.

Vertical Shifts: f(x) + c

f(x) + c shifts the graph up by c units.

f(x) - c shifts the graph down by c units.

The output changes, but the input doesn't. Every y-value increases by c.

Example: f(x) = x². The vertex is at (0, 0).

g(x) = x² + 3. The vertex is at (0, 3). Shifted up 3 units.

h(x) = x² - 5. The vertex is at (0, -5). Shifted down 5 units.

The shape is identical. Only the position changes.

Horizontal Shifts: f(x - c)

f(x - c) shifts the graph right by c units.

f(x + c) shifts the graph left by c units.

This is counterintuitive. Subtracting shifts right. Adding shifts left.

Why? Because you're changing the input, not the output.

Think about f(x - 2). To get the same output you used to get at x = 0, you now need x = 2. The function has to "catch up" by moving right.

Example: f(x) = x². The vertex is at (0, 0).

g(x) = (x - 2)². The vertex is at (2, 0). Shifted right 2 units.

h(x) = (x + 3)². The vertex is at (-3, 0). Shifted left 3 units.

The sign is opposite to the direction. Minus means right. Plus means left.

Combining Vertical and Horizontal Shifts

g(x) = (x - 2)² + 3.

Start with f(x) = x². Vertex at (0, 0).

The (x - 2) shifts right 2 units. Vertex moves to (2, 0).

The +3 shifts up 3 units. Vertex moves to (2, 3).

The graph of g is the graph of f shifted right 2, up 3.

Order doesn't matter. You can shift vertically first or horizontally first. The result is the same.

Vertical Stretches and Compressions: c·f(x)

c·f(x) multiplies all y-values by c.

If c > 1, the graph stretches vertically. It gets taller.

If 0 < c < 1, the graph compresses vertically. It gets shorter.

Example: f(x) = x². The point (2, 4) is on the graph.

g(x) = 2·f(x) = 2x². The point (2, 8) is on the graph. The y-value doubled.

h(x) = 0.5·f(x) = 0.5x². The point (2, 2) is on the graph. The y-value halved.

The x-values don't change. The y-values scale.

Horizontal Stretches and Compressions: f(cx)

f(cx) changes the x-scale.

If c > 1, the graph compresses horizontally. It gets narrower.

If 0 < c < 1, the graph stretches horizontally. It gets wider.

Again, this is counterintuitive. Multiplying x by a larger number makes the graph narrower, not wider.

Why? Because f(2x) reaches the same output in half the distance. To get the output that f used to produce at x = 2, you now only need x = 1. The graph compresses horizontally.

Example: f(x) = x². The point (2, 4) is on the graph.

g(x) = f(2x) = (2x)² = 4x². The point (1, 4) is on the graph. The x-value halved.

h(x) = f(0.5x) = (0.5x)² = 0.25x². The point (4, 4) is on the graph. The x-value doubled.

Rule of thumb: Larger coefficient inside the function = narrower graph. Smaller coefficient inside = wider graph.

Vertical Reflection: -f(x)

-f(x) flips the graph upside down across the x-axis.

Every positive y-value becomes negative. Every negative y-value becomes positive.

Example: f(x) = x². The graph opens upward.

g(x) = -x². The graph opens downward.

The point (2, 4) on f becomes (2, -4) on g.

Horizontal Reflection: f(-x)

f(-x) flips the graph left-to-right across the y-axis.

Every point (a, b) becomes (-a, b).

Example: f(x) = x³. The graph rises to the right, falls to the left.

g(x) = (-x)³ = -x³. The graph falls to the right, rises to the left.

The point (2, 8) on f becomes (-2, 8) on g.

(Note: (-x)³ = -x³ because the exponent is odd. For even exponents, f(-x) = f(x).)

Order of Transformations

When multiple transformations are applied, order matters.

General form: a·f(b(x - h)) + k.

• h: horizontal shift
• k: vertical shift
• a: vertical stretch/compression and reflection
• b: horizontal stretch/compression and reflection

Order of operations (inside to outside):

1. Horizontal shift: x - h
2. Horizontal stretch/compression: b(x - h)
3. Vertical stretch/compression: a·f(...)
4. Vertical shift: + k

Example: g(x) = -2(x - 3)² + 5.

Start with f(x) = x².

1. Horizontal shift right 3: f(x - 3) = (x - 3)².
2. Vertical stretch by 2: 2f(x - 3) = 2(x - 3)².
3. Vertical reflection: -2(x - 3)².
4. Vertical shift up 5: -2(x - 3)² + 5.

The vertex of f(x) = x² is (0, 0). After transformations, the vertex of g is (3, 5).

Reading Transformations from the Equation

Given g(x) = 3f(2(x + 1)) - 4, what transformations were applied to f?

Inside the function: 2(x + 1).

x + 1 means shift left 1. (Remember: plus means left.)

2 means horizontal compression by a factor of 2 (narrower).

Outside the function: 3(...) - 4.

3 means vertical stretch by a factor of 3 (taller).

-4 means shift down 4.

Transformations:

• Shift left 1
• Horizontal compression by 2
• Vertical stretch by 3
• Shift down 4

Why Horizontal Shifts Are Counterintuitive

f(x - 2) shifts right. f(x + 2) shifts left. This trips people up.

The key: you're replacing x with (x - c). To get the output that used to occur at x = a, you now need x - c = a, so x = a + c.

The graph shifts in the direction that makes the new input equal the old input.

Example: f(0) gives the y-intercept. For f(x - 2), you get that same output at x = 2. The graph shifted right.

Symmetry and Transformations

Some functions are symmetric.

Even function: f(-x) = f(x). Symmetric across the y-axis.

Example: f(x) = x², f(x) = cos(x).

Odd function: f(-x) = -f(x). Symmetric across the origin (180° rotational symmetry).

Example: f(x) = x³, f(x) = sin(x).

Transformations can break or preserve symmetry.

If f is even, then f(x - h) is not even (unless h = 0). Horizontal shifts break y-axis symmetry.

But f(x) + k is still even. Vertical shifts preserve y-axis symmetry.

Transformations of Standard Functions

Linear: f(x) = x.

Vertical stretch by m, vertical shift by b: g(x) = mx + b. This is the slope-intercept form of a line.

Quadratic: f(x) = x².

Standard form: g(x) = a(x - h)² + k. Vertex at (h, k).

Square root: f(x) = √x.

g(x) = √(x - h) + k. Starting point (where the graph begins) at (h, k).

Absolute value: f(x) = |x|.

g(x) = a|x - h| + k. Vertex at (h, k).

Reciprocal: f(x) = 1/x.

g(x) = a/(x - h) + k. Vertical asymptote at x = h, horizontal asymptote at y = k.

Transformations and Vertex Form

For parabolas, vertex form makes transformations explicit.

f(x) = a(x - h)² + k.

• Vertex: (h, k)
• a > 0: opens upward
• a < 0: opens downward
• |a| > 1: narrower
• |a| < 1: wider

Example: f(x) = -2(x + 3)² + 7.

Vertex: (-3, 7). Opens downward. Narrow (|a| = 2 > 1).

Compare to standard form: f(x) = ax² + bx + c. The transformations are hidden. You'd have to complete the square to extract them.

Vertex form makes the geometry transparent.

Practical Applications

Physics: A projectile's height is h(t) = -16(t - t₀)² + h₀. The vertex (t₀, h₀) gives the time and height of the maximum.

Economics: A profit function might be P(x) = -a(x - x₀)² + P₀. The vertex (x₀, P₀) gives the optimal quantity and maximum profit.

Signal processing: Shifting and scaling waveforms.

Computer graphics: Transforming objects in space.

Transformations aren't abstract. They're how you encode position, orientation, and scale.

Transformations vs. Plotting Points

You could plot g(x) = 2(x - 3)² + 1 by choosing x-values, computing g(x), and plotting points.

Or you could recognize: this is the parabola y = x², stretched vertically by 2, shifted right 3, shifted up 1. Vertex at (3, 1). Opens upward. Narrow.

You see the shape immediately, without computation.

This is the power of transformations: you're reading structure from the equation, not calculating outputs.

Common Mistakes

Mistake 1: Getting horizontal shift direction backwards.

f(x - 3) shifts right, not left. The sign is opposite to the direction.

Mistake 2: Confusing vertical and horizontal stretch.

2f(x) is vertical stretch. f(2x) is horizontal compression.

Mistake 3: Applying transformations in the wrong order.

Horizontal operations (inside f) happen first. Vertical operations (outside f) happen second.

Mistake 4: Forgetting that f(cx) compresses for c > 1.

Larger coefficient → narrower graph. It's counterintuitive, but consistent.

Transformations as Function Arithmetic

You can think of transformations as operations on functions.

f(x) + c: add a constant to the function.

c·f(x): multiply the function by a constant.

f(x + c): shift the input by a constant.

f(cx): scale the input by a constant.

This generalizes. You can add two functions: (f + g)(x) = f(x) + g(x). You can multiply them: (f·g)(x) = f(x)·g(x).

Transformations are the simplest cases: operations involving constants.

Transformations and Composition

A transformation is a function applied to a function.

Let T(f(x)) = f(x) + 3. This is a transformation: shift up by 3.

Let S(f(x)) = f(x - 2). This is a transformation: shift right by 2.

Composing transformations applies them sequentially.

(T ∘ S)(f(x)) = T(S(f(x))) = T(f(x - 2)) = f(x - 2) + 3.

Shift right 2, then shift up 3.

Transformations compose. You can build complex transformations from simple ones.

Inverse Transformations

Every transformation has an inverse.

f(x) + 3 shifts up 3. The inverse shifts down 3: f(x) - 3.

f(x - 2) shifts right 2. The inverse shifts left 2: f(x + 2).

2f(x) stretches by 2. The inverse compresses by 2: f(x)/2.

-f(x) reflects. The inverse reflects again: -(-f(x)) = f(x).

To undo a transformation, apply the inverse transformation.

Transformations and the Graph Library

You don't need to memorize the graph of every function.

You need to know the graphs of a few basic functions:

• f(x) = x (line)
• f(x) = x² (parabola)
• f(x) = x³ (cubic)
• f(x) = √x (square root)
• f(x) = 1/x (hyperbola)
• f(x) = |x| (absolute value)
• f(x) = e^x (exponential)
• f(x) = ln(x) (logarithm)
• f(x) = sin(x), cos(x) (trig)

Once you know these, you can generate any transformed version by applying shifts, stretches, and reflections.

g(x) = -2e^(x + 1) - 3?

Start with e^x. Shift left 1. Stretch vertically by 2. Reflect vertically. Shift down 3. Done.

The graph library + transformations = every function you'll encounter.

Seeing Structure

When you look at g(x) = 3√(x - 2) + 1, you don't see a formula to evaluate.

You see: square root function, shifted right 2, stretched vertically by 3, shifted up 1.

You see structure. You see geometry encoded in algebra.

That's the payoff of understanding transformations. You stop computing and start seeing.

Part 3 of the Precalculus series.

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