Stretches of Graphs

Welcome, students! In this lesson, you will learn how stretches of graphs change the shape of a function and how to describe those changes using function notation and transformations. Stretches are one of the most important ideas in the study of functions because they let you model real situations where values grow faster, slower, or by different amounts depending on the input 📈.

What you will learn

What a stretch of a graph means in function language
The difference between vertical and horizontal stretches
How stretches affect common functions such as polynomial, rational, exponential, and logarithmic graphs
How to use stretches with other transformations like shifts and reflections
How stretches connect to IB Mathematics: Analysis and Approaches HL function reasoning

Keep in mind: transformations are not just about drawing pretty graphs. They help you describe changes in patterns, compare models, and solve problems in context.

1. What is a stretch?

A stretch changes the size of a graph in a chosen direction without changing the basic type of function. If a graph is stretched, one coordinate direction is pushed farther away from the axis, while the other direction may stay the same.

For a function $y=f(x)$, a vertical stretch by a factor of $a$ gives the graph of

$$y=af(x).$$

If $a>1$, the graph becomes taller. If $0<a<1$, the graph becomes shorter, which is often called a vertical compression.

For example, if $f(x)=x^2$, then $y=3x^2$ is a vertical stretch by factor $3$. Compared with $y=x^2$, the parabola rises more quickly. On the other hand, $y=\frac{1}{2}x^2$ is shorter and wider-looking.

A horizontal stretch by factor $k$ is written as

$$y=f\left(\frac{x}{k}\right).$$

If $k>1$, the graph becomes wider. If $0<k<1$, the graph becomes narrower. Horizontal stretches can feel less intuitive because the change happens inside the function.

Important idea: a vertical stretch multiplies output values, while a horizontal stretch changes input values. That difference matters a lot in IB exams.

2. Vertical stretches in function notation

Suppose students is given a function $f(x)$ and asked to transform it into $y=af(x)$. This means every point $(x,y)$ on the graph of $y=f(x)$ becomes $(x,ay)$ on the new graph.

So the $x$-coordinates stay the same, but the $y$-coordinates are multiplied by $a$.

Example 1

If $f(x)=x^3$, then $g(x)=2f(x)=2x^3$ is a vertical stretch by factor $2$.

The point $(1,1)$ on $y=x^3$ becomes $(1,2)$.
The point $(-1,-1)$ becomes $(-1,-2)$.

This stretch makes the graph steeper away from the origin.

Example 2

If $f(x)=\sqrt{x}$, then $g(x)=\frac{1}{3}f(x)=\frac{1}{3}\sqrt{x}$ is a vertical compression by factor $\frac{1}{3}$.

The graph keeps the same domain, $x\ge 0$.
Every output is one third of the original output.

Vertical stretches are especially useful when comparing models. For instance, two temperatures may follow the same pattern, but one city may consistently record values three times larger than another due to unit scaling.

3. Horizontal stretches and why they look reversed

Horizontal stretches are often trickier because they happen inside the function.

If $g(x)=f\left(\frac{x}{k}\right)$ with $k>1$, then the graph is stretched horizontally by factor $k$.

This means every point $(x,y)$ on $y=f(x)$ becomes $(kx,y)$ on the new graph.

So the $y$-coordinates stay the same, but the $x$-coordinates are multiplied by $k$.

Example 3

If $f(x)=x^2$, then

$$g(x)=f\left(\frac{x}{2}\right)=\left(\frac{x}{2}\right)^2$$

is a horizontal stretch by factor $2$.

The point $(1,1)$ becomes $(2,1)$.
The point $(2,4)$ becomes $(4,4)$.

The graph looks wider because it takes larger $x$-values to produce the same $y$-value.

A common mistake is to think that $f(2x)$ is a horizontal stretch by factor $2$. In fact, $f(2x)$ causes a horizontal compression by factor $\frac{1}{2}$.

Key rule

$y=af(x)$ gives a vertical stretch by factor $a$
$y=f\left(\frac{x}{k}\right)$ gives a horizontal stretch by factor $k$
$y=f(kx)$ gives a horizontal compression by factor $\frac{1}{k}$ when $k>1$

This inverse relationship is one of the most tested ideas in function transformation questions.

4. Stretches with common families of functions

Stretches appear across the main function types in the IB syllabus.

Polynomial functions

For $f(x)=x^2$, $y=3x^2$ is vertically stretched, while $y=\left(\frac{x}{3}\right)^2$ is horizontally stretched.

Polynomial graphs are smooth and their shapes can change noticeably under stretch. A larger vertical factor makes the graph rise or fall more rapidly. For higher-degree polynomials, stretches can make the graph appear much steeper near the origin.

Rational functions

For a rational function such as $f(x)=\frac{1}{x}$, a vertical stretch gives

$$y=\frac{2}{x}.$$

This makes the branches move farther from the axes for the same $x$-values.

A horizontal stretch gives

$$y=\frac{1}{x/2}=\frac{2}{x}$$

which happens to produce the same algebraic expression in this case. This is a useful reminder that different transformation paths can lead to the same simplified formula.

Exponential functions

For $f(x)=2^x$, the graph of $y=3\cdot 2^x$ is a vertical stretch by factor $3$.

This means the exponential growth is faster at every $x$-value, because all outputs are multiplied by $3$.

A horizontal stretch such as

$$y=2^{x/2}$$

changes how quickly the input grows. Since $x$ is divided by $2$, the graph grows more slowly along the horizontal axis.

Logarithmic functions

For $f(x)=\log x$, a vertical stretch gives

$$y=2\log x.$$

The graph becomes taller but keeps the same domain $x>0$.

A horizontal stretch changes how quickly the graph moves away from its vertical asymptote. For example,

$$y=\log\left(\frac{x}{2}\right)$$

shifts the graph’s input scale, making it take larger $x$-values to achieve the same output.

5. Combining stretches with other transformations

In IB Mathematics, graphs are often transformed by several changes at once. A function might be stretched, shifted, and reflected in a single expression.

For example,

$$y=-2f(x-3)$$

means:

a horizontal shift right by $3$
a vertical stretch by factor $2$
a reflection in the $x$-axis because of the negative sign

The order matters when interpreting the graph, but the expression gives all the needed information.

Example 4

If $f(x)=x^2$, then

$$g(x)=-3(x-1)^2+4$$

has these transformations:

shift right by $1$
vertical stretch by factor $3$
reflection in the $x$-axis
shift up by $4$

So the vertex moves from $(0,0)$ to $(1,4)$, and the parabola opens downward more sharply than the original.

This kind of reasoning is exactly what you need when sketching graphs from formulas or writing formulas from graphs.

6. How stretches help with equations and graphs

Stretches are not just visual. They can help solve equations and compare functions.

If two graphs are related by a stretch, their key points and intercepts may be connected in a predictable way. This can help with estimating solutions, checking answers, or building a model from data.

For example, if $y=f(x)$ and $y=2f(x)$, then any $x$-value where $f(x)=0$ is also a zero of $2f(x)$, because

$$2f(x)=0$$

occurs exactly when

$$f(x)=0.$$

So vertical stretches do not change the roots of a function.

Horizontal stretches can change where important features occur along the $x$-axis. If a feature appears at $x=c$ on $y=f(x)$, then on $y=f\left(\frac{x}{k}\right)$ it appears at $x=kc$.

This matters in applications such as sound waves, motion graphs, and scaling time in models. If a process takes twice as long, a horizontal stretch may be the correct way to represent it.

Conclusion

Stretches of graphs are a core part of function transformations in IB Mathematics: Analysis and Approaches HL. A vertical stretch changes output values using $y=af(x)$, while a horizontal stretch changes input scale using $y=f\left(\frac{x}{k}\right)$. These transformations work across polynomial, rational, exponential, and logarithmic functions, and they often appear together with shifts and reflections.

For students, the key skill is to read a function expression carefully and decide whether the change is affecting $x$-values or $y$-values. That understanding helps with graph sketching, modeling, and solving problems involving function behavior.

Study Notes

A vertical stretch by factor $a$ is written as $y=af(x)$.
A horizontal stretch by factor $k$ is written as $y=f\left(\frac{x}{k}\right)$.
Vertical stretches multiply $y$-values; horizontal stretches multiply $x$-values.
If $a>1$, $y=af(x)$ is taller; if $0<a<1$, it is a vertical compression.
If $k>1$, $y=f\left(\frac{x}{k}\right)$ is wider; if $0<k<1$, it is a horizontal compression.
$f(2x)$ is a horizontal compression by factor $\frac{1}{2}$, not a stretch.
Stretches apply to polynomial, rational, exponential, and logarithmic functions.
Vertical stretches do not change the zeros of a function.
Horizontal stretches change where features appear along the $x$-axis.
Stretches are often combined with shifts and reflections in transformation questions.
In IB problems, always identify whether the transformation acts on the outside or inside of $f(x)$.