- Stretches and compressions are dilations applied to the graph of a function.
- They change the scale of the graph either vertically (changing $y$-values) or horizontally (changing $x$-values).
- Understanding the difference between these two is essential, because they look similar but work in opposite directions.
Dilations Change Scale Without Changing The Overall Shape
Definition
Dilation
A transformation that multiplies all distances from a center point by the same scale factor, producing a similar image.
- A dilation makes a graph look "stretched" or "squashed," but the curve remains the same kind of curve (for example, a parabola stays a parabola).
- In this article you will cover these function transformations:
- Vertical dilation: $y = a f(x)$
- Horizontal dilation: $y = f(ax)$
- They are easy to confuse because the symbol $a$ appears in both, but the effect is different.
Vertical Dilations Scale The Output: $g(x)=a f(x)$
- When you create a new function $$g(x)=a f(x)$$ you multiply every output (every $y$-value) of $f$ by $a$.
- That means each point $(x,y)$ on $y=f(x)$ moves to $$(x, ay)$$ on $y=g(x)$.
- What stays the same:
- The $x$-coordinates of all points stay the same.
- The $x$-intercepts stay the same (because $y=0$ stays $0$ after multiplying).
- What changes:
- All $y$-coordinates are multiplied by $a$.
- The graph is dilated parallel to the $y$-axis.
Definition
Vertical dilation
The transformation $y=a f(x)$, where all $y$-values are multiplied by $a$. It is a dilation with scale factor $a$ parallel to the $y$-axis.
How The Value Of $a$ Affects The Graph
- If $a>1$, the graph is a vertical stretch (points move further from the $x$-axis).
- If $0<a<1$, the graph is a vertical compression (points move closer to the $x$-axis).
Example
- A function $f$ passes through $(1,-1)$.
- For $g(x)=2f(x)$, the new point is $(1,2\cdot(-1))=(1,-2)$.
- For $g(x)=\tfrac12 f(x)$, the new point is $(1,\tfrac12\cdot(-1))=(1,-\tfrac12)$.
Example
Using A Table Of Points (From A Plotted Curve)
Suppose $f(x)$ is defined by the points
$$(-4,-5),(-3,-2),(-2,0),(-1,1),(0,0),(1,-1),(4,1),(5,3)$$
For $g_1(x)=2f(x)$, double each $y$-coordinate:
- $(-4,-5)\to(-4,-10)$
- $(-3,-2)\to(-3,-4)$
- $(-2,0)\to(-2,0)$
- $(-1,1)\to(-1,2)$
- $(0,0)\to(0,0)$
- $(1,-1)\to(1,-2)$
- $(4,1)\to(4,2)$
- $(5,3)\to(5,6)$
For $g_2(x)=\tfrac12 f(x)$, halve each $y$-coordinate:
- $(-4,-5)\to(-4,-2.5)$
- $(-3,-2)\to(-3,-1)$
- $(-2,0)\to(-2,0)$
- $(-1,1)\to(-1,0.5)$
- $(0,0)\to(0,0)$
- $(1,-1)\to(1,-0.5)$
- $(4,1)\to(4,0.5)$
- $(5,3)\to(5,1.5)$
Horizontal Dilations Scale The Input: $h(x)=f(ax)$
- When you create $$h(x)=f(ax),$$ you change the input before the function acts.
- A quick way to track points is to start from a point on $f$.
- If $(x,y)$ lies on $y=f(x)$ (so $y=f(x)$), then on $y=f(ax)$ you need an $x$-value that makes $ax$ equal to the original input.
- So the point mapping is: $$(x,y) \text{ on } f \quad\longrightarrow\quad \left(\frac{x}{a},y\right) \text{ on } f(ax)$$
- That means the graph is dilated parallel to the $x$-axis with scale factor $\frac{1}{a}$.
Definition
Horizontal dilation
The transformation $y=f(ax)$, where $x$-values are scaled by a factor $\frac{1}{a}$. It is a dilation parallel to the $x$-axis.
How The Value Of $a$ Affects The Graph
- If $a>1$, then $\frac{1}{a}<1$, so the graph is horizontally compressed.
- If $0<a<1$, then $\frac{1}{a}>1$, so the graph is horizontally stretched.
Analogy
- Think of $f(ax)$ like playing a video faster.
- If you speed up by factor 2, the same "events" happen in half the time, so the graph squeezes toward the $y$-axis.
Example
Let $f(x)=x^2$.
- For $2f(x)=2x^2$, a point like $(1,1)$ becomes $(1,2)$, it moves vertically.
- For $f(2x)=(2x)^2=4x^2$, the $x$-coordinate is affected.
- For instance, the point $(2,4)$ on $y=x^2$ becomes $(1,4)$ on $y=f(2x)$, because $\frac{2}{2}=1$.
This illustrates the key idea: $f(2x)$ compresses horizontally by factor $\tfrac12$.
Tip
- To sketch $y=f(ax)$, you can often take key $x$-coordinates from the original graph and divide them by $a$.
- The $y$-values stay the same.
Exam technique
General Rules You Should Memorize
Summary For $y=a f(x)$
- A point $(x,y)$ becomes $(x,ay)$.
- $a>1$: vertical stretch by factor $a$
- $0<a<1$: vertical compression by factor $a$
Summary For $y=f(ax)$
- A point $(x,y)$ becomes $\left(\tfrac{x}{a},y\right)$.
- $a>1$: horizontal compression by factor $\tfrac{1}{a}$
- $0<a<1$: horizontal stretch by factor $\tfrac{1}{a}$
Many students find it helpful to say it out loud: in $f(ax)$ the "$a$" is inside, so the effect on the graph is the reciprocal outside, a horizontal scale factor of $\frac{1}{a}$.
Connecting Function Dilations To Geometric Dilations
- In geometry, a dilation of scale factor $r$ multiplies lengths by $r$.
- If $r>1$, it is an enlargement.
- If $0<r<1$, it is a reduction.
- However, position matters only when a center of dilation is specified.
Definition
Center of dilation
A fixed point $C$ such that each point of the image lies on the line from $C$ through the original point, and its distance from $C$ is multiplied by the scale factor.
For geometric figures:
- With scale factor only, the enlarged/reduced figure can be placed in many positions (position is arbitrary).
- With scale factor and center, the image is fixed by the rule "same direction from the center, scaled distance."
This is exactly what happens when you dilate from the origin: every point is moved along a ray from $(0,0)$.
Case study
- If you dilate a figure from the origin by factor 3, each vertex $(x,y)$ moves to $(3x,3y)$.
- The image is three times as far from the origin in the same direction, matching the "aligned points" method used when constructing an enlargement from a chosen point.
Negative Scale Factors (Extension Idea)
- Scale factors are often given with $0<r<1$ or $r>1$, but a scale factor can also be negative.
- For functions, $y=a f(x)$ with $a<0$ combines a vertical dilation by $|a|$ with a reflection in the $x$-axis (because all $y$-values change sign).
- For geometric dilation about a center, a negative factor places points on the opposite side of the center (a half-turn effect) as well as scaling distances.
Common Mistake
- If $a$ is negative, do not describe the transformation as "just a stretch."
- You must also mention the relevant reflection (and for geometric dilation, the reversal through the center).
Common Sketching Workflow
When you are given a graph of $y=f(x)$ and asked to sketch a transformed graph:
- Identify which type you have: $a f(x)$ (vertical) or $f(ax)$ (horizontal).
- Choose a few clear points (intercepts, turning points, labeled points).
- Transform each point using the mapping rule.
- Sketch the new curve smoothly through the transformed points.
Exam technique
- For exam sketching, you usually need 4 to 6 well-chosen points, not dozens.
- Prioritize: intercepts, maximum/minimum points, and any clearly given coordinates.
Self review
- If $g(x)=0.3f(x)$, is this a stretch or compression, and in which direction?
- If $h(x)=f(5x)$, what is the horizontal scale factor? Does the graph get wider or narrower?
- A point $(6,-2)$ lies on $f$. Where does it go on $y=f(\tfrac12 x)$?
