Stretches and Compressions: Scaling Outputs and Inputs

Vertical Scaling: a f(x) (Stretch/Compression of Outputs)

Vertical scaling multiplies the output of a function by a constant a. If the original function is y = f(x), the scaled function is:

y = a f(x)

This changes how far points on the graph sit above or below the x-axis without changing their x-coordinates.

Magnitude: stretch vs compression

• If |a| > 1, outputs get larger in magnitude: a vertical stretch.
• If 0 < |a| < 1, outputs get smaller in magnitude: a vertical compression.
• If a = 1, no change.
• If a = 0, every output becomes 0, so the graph becomes the x-axis (y = 0).

Sign: reflection across the x-axis

• If a > 0, the graph keeps its up/down orientation.
• If a < 0, outputs change sign, so the graph is reflected across the x-axis (and also scaled by |a|).

Predicting changes using function notation

Start with a known output: if f(2) = -3, then for g(x) = 4 f(x):

• g(2) = 4 f(2) = 4(-3) = -12 (same input, output multiplied by 4)

If h(x) = -0.5 f(x), then:

• h(2) = -0.5(-3) = 1.5 (compression by 1/2 and reflection)

Quick point rule for a f(x)

If a point (x, y) lies on y = f(x), then the corresponding point on y = a f(x) is:

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(x, y)  →  (x, a y)

Only the y-coordinate is multiplied by a.

Horizontal Scaling: f(bx) (Stretch/Compression of Inputs)

Horizontal scaling multiplies the input by a constant b inside the function:

y = f(bx)

This changes where the graph sits left/right by changing which x-values produce the same outputs.

The reciprocal effect (most common mistake)

Horizontal scaling works in the opposite way you might first guess:

• If |b| > 1, the graph is horizontally compressed by a factor of 1/|b|.
• If 0 < |b| < 1, the graph is horizontally stretched by a factor of 1/|b| (which is > 1).

Why “reciprocal”? Because to get the same original input value into f, you must use an x-value that is divided by b.

Sign: reflection across the y-axis

• If b > 0, left/right orientation stays the same.
• If b < 0, the graph is reflected across the y-axis (and scaled horizontally by 1/|b|).

Predicting changes using function notation

Suppose f(3) = 5. Define g(x) = f(2x). To find an x-value where g(x) = 5, set the inside equal to 3:

• Need 2x = 3 so x = 3/2.
• Then g(3/2) = f(2·3/2) = f(3) = 5.

This shows the x-coordinate is divided by 2, so the graph moves closer to the y-axis (compression).

Quick point rule for f(bx)

If a point (x, y) lies on y = f(x), then the corresponding point on y = f(bx) is:

(x, y)  →  (x/b, y)

Only the x-coordinate changes, and it is divided by b (including sign).

Point-Mapping Rules: Transform Key Points Efficiently

When you know key points of y = f(x) (intercepts, vertices, corners, endpoints), scaling is often fastest by mapping points rather than re-plotting from scratch.

Separate rules

Combined scaling: y = a f(bx)

Apply both effects:

(x, y)  →  (x/b, a y)

Tip: Map x using b (divide by b), map y using a (multiply by a).

Step-by-step example (mapping key points)

Assume the graph of y = f(x) includes these key points:

• (-2, 1), (0, -3), (4, 2)

Transform to y = -2 f(3x).

1. Horizontal part: f(3x) means x → x/3.
2. Vertical part: -2 f(...) means y → -2y.
3. Map each point using (x, y) → (x/3, -2y):

• (-2, 1) → (-2/3, -2)
• (0, -3) → (0, 6)
• (4, 2) → (4/3, -4)

Plot the transformed points and connect them with the same basic shape as the original graph.

Comparing Scaling Effects Across Function Families

The algebraic form changes in a predictable way, but the visual meaning of scaling can look different depending on the family.

Lines: scaling changes slope magnitude

Let f(x) = mx + c. Then:

• a f(x) = a(mx + c) = (am)x + (ac) so the slope becomes am (steeper if |a|>1, flatter if 0<|a|<1), and the y-intercept becomes ac.
• f(bx) = m(bx) + c = (mb)x + c so the slope becomes mb. Horizontal compression by b also makes the line appear steeper by the same factor |b|.

Example: If f(x)=2x-1, then f(3x)=6x-1 (slope triples), while 0.5f(x)=x-0.5 (slope halves).

Parabolas: scaling changes “steepness” (narrow/wide)

Let f(x)=x^2.

• a f(x)=a x^2: larger |a| makes the parabola narrower/steeper; smaller |a| makes it wider/flatter; negative a reflects it to open downward.
• f(bx)=(bx)^2=b^2 x^2: horizontal scaling by b produces a vertical factor of b^2 in the equation. Visually it is still a horizontal compression/stretch, but algebraically it looks like a vertical change because of the square.

Example: y=f(2x)=(2x)^2=4x^2. The graph is horizontally compressed by 1/2, and it also appears much narrower (equivalently, vertically stretched by 4 compared to x^2).

Absolute value: scaling changes the “V” opening

Let f(x)=|x|.

• a|x| makes the V steeper (|a|>1) or flatter (0<|a|<1); negative a flips it upside down.
• |bx|=|b||x| shows that horizontal scaling becomes a vertical factor of |b| in the formula, even though the transformation is horizontal.

Square root: horizontal scaling changes how quickly it grows

Let f(x)=√x.

• a√x scales outputs directly.
• √(bx) compresses or stretches horizontally; for b>1, the curve reaches the same y-values at smaller x-values.

Practice: Identify the Transformation from an Equation

For each, describe the vertical scaling (factor and reflection if any) and horizontal scaling (factor and reflection if any) relative to y=f(x).

1. y = 3 f(x)
2. y = -0.25 f(x)
3. y = f(5x)
4. y = f(0.2x)
5. y = -2 f(-3x)
6. y = 0.5 f(4x)

Practice check (answers)

• 1) Vertical stretch by 3.
• 2) Vertical compression by 1/4 and reflection across x-axis.
• 3) Horizontal compression by factor 1/5.
• 4) Horizontal stretch by factor 5 (since 1/0.2 = 5).
• 5) Vertical stretch by 2 with reflection across x-axis; horizontal compression by 1/3 with reflection across y-axis.
• 6) Vertical compression by 1/2; horizontal compression by 1/4.

Practice: Construct an Equation from a Described Scaling

Write the transformed function in terms of f.

1. Take y=f(x), reflect across the x-axis and vertically stretch by 4.
2. Take y=f(x), horizontally stretch by factor 3.
3. Take y=f(x), horizontally compress by factor 2 and vertically compress by factor 1/5.
4. Take y=f(x), reflect across the y-axis and horizontally stretch by factor 6.
5. Take y=f(x), reflect across both axes and horizontally compress by factor 4.

Practice check (answers)

• 1) y = -4 f(x)
• 2) Horizontal stretch by 3 means multiply input by 1/3: y = f((1/3)x) or y=f(x/3).
• 3) Horizontal compress by 2 means f(2x); vertical compress by 1/5 means multiply outputs by 1/5: y = (1/5) f(2x).
• 4) Reflect across y-axis means x→-x. Horizontal stretch by 6 means use x/6 inside: y = f(-(x/6)) or y=f(-x/6).
• 5) Horizontal compress by 4: f(4x). Reflect across y-axis too: f(-4x). Reflect across x-axis: multiply by -1. Final: y = -f(-4x).

Mixed Practice: Use Key Points to Build the Transformed Graph

Suppose y=f(x) passes through the points (-4,2), (-1,-3), (2,1), and (5,0). Map these points to the transformed graph.

1. y = 2 f(x)
2. y = f(0.5x)
3. y = -3 f(2x)

Practice check (mapped points)

• 1) Use (x,y)→(x,2y): (-4,4), (-1,-6), (2,2), (5,0).
• 2) Use (x,y)→(x/0.5,y)=(2x,y): (-8,2), (-2,-3), (4,1), (10,0).
• 3) Use (x,y)→(x/2,-3y): (-2,-6), (-1/2,9), (1,-3), (5/2,0).