Stretches of Graphs 📈

Functions help us describe patterns, and graphs help us see them. In this lesson, students, you will learn how a graph can be stretched vertically or horizontally, how these transformations change the appearance of a function, and how to recognize them in real-world situations. Stretching is one of the most important ideas in function transformations because it changes the size of a graph without changing its overall shape. That makes it useful for comparing models of growth, decay, and change in IB Mathematics Analysis and Approaches SL.

What is a stretch?

A stretch changes a graph by pulling it away from or pushing it toward an axis. The graph keeps the same general shape, but it becomes taller, wider, or narrower. This is different from a translation, which moves a graph without changing its size, and different from a reflection, which flips a graph.

There are two main kinds of stretches:

Vertical stretch: changes the $y$-values of the graph.
Horizontal stretch: changes the $x$-values of the graph.

If a function is $f(x)$, then a vertical stretch by a factor of $a$ gives the new function $y = af(x)$. A horizontal stretch by a factor of $a$ gives the new function $y = f\left(\frac{x}{a}\right)$. These two formulas are extremely important in IB function work.

A common point of confusion is that the number works differently in the two formulas. For a vertical stretch, the graph is multiplied directly by the factor. For a horizontal stretch, the factor appears inside the function with $x$ divided by that number. That means the graph stretches in the opposite way you might first expect.

Vertical stretches: making graphs taller or shorter

A vertical stretch multiplies every output of the function by the same number. If the original function is $f(x)$, then the transformed function is $y = af(x)$.

If $a > 1$, the graph is stretched away from the $x$-axis.
If $0 < a < 1$, the graph is compressed toward the $x$-axis.
If $a < 0$, the graph is also reflected in the $x$-axis, because the factor is negative.

For example, if $f(x) = x^2$, then $y = 2x^2$ is a vertical stretch by factor $2$. Every $y$-value becomes twice as large. The point $(1,1)$ becomes $(1,2)$, and $(2,4)$ becomes $(2,8)$.

This matters in modeling. Suppose a company’s profit is described by $P(x)$, where $x$ is the number of items sold. If a new tax or bonus system doubles the profit for each sale, the new model could be $2P(x)$. The shape of the graph stays the same, but the height changes.

Vertical stretches are easy to see on graphs of familiar functions. For a line $f(x) = x$, the graph of $y = 3x$ is steeper than the original. For a quadratic function, the parabola becomes narrower when stretched vertically. For an exponential function like $f(x) = 2^x$, the graph grows faster when multiplied by a number greater than $1$.

Horizontal stretches: making graphs wider or narrower

Horizontal stretches are a little trickier because they affect the input values, not the outputs. If the function is $f(x)$, then a horizontal stretch by factor $a$ is written as $y = f\left(\frac{x}{a}\right)$.

If $a > 1$, the graph is stretched away from the $y$-axis.
If $0 < a < 1$, the graph is compressed toward the $y$-axis.

Why does this happen? If the graph is $y = f\left(\frac{x}{a}\right)$, then to get the same output as before, you must use an $x$-value that is $a$ times bigger. So the points move farther apart horizontally.

For example, if $f(x) = x^2$, then $y = f\left(\frac{x}{2}\right) = \left(\frac{x}{2}\right)^2$ is a horizontal stretch by factor $2$. A point that was at $(1,1)$ on $f(x)$ is now at $(2,1)$ on the new graph. The graph becomes wider.

This is especially useful when studying functions that model time. If a temperature change originally happens over $3$ hours but now happens over $6$ hours, the graph may be stretched horizontally. In that case, the same output values are reached more slowly.

A very important fact for students to remember is that horizontal stretching changes the graph in the opposite direction inside the function. This is one of the most common exam mistakes in transformation questions.

Comparing vertical and horizontal stretches

Vertical and horizontal stretches both change the size of a graph, but they do it in different ways.

Vertical stretch: $y = af(x)$ changes the outputs.
Horizontal stretch: $y = f\left(\frac{x}{a}\right)$ changes the inputs.

A helpful way to think about it is this: vertical stretches change how high a graph goes, while horizontal stretches change how far across it spreads.

Consider the function $f(x) = x^2$.

$y = 2x^2$ is vertically stretched by factor $2$.
$y = \left(\frac{x}{2}\right)^2$ is horizontally stretched by factor $2$.

These graphs are not the same. One becomes narrower and taller, while the other becomes wider. In many tasks, you may need to identify which transformation has happened from a graph or from an equation. Looking carefully at the coordinates of key points, such as intercepts or turning points, helps you decide.

Stretches in graph representations and function language

In IB Mathematics Analysis and Approaches SL, you need to move easily between an equation, a graph, and a description in words. Stretches are a perfect example of this skill.

If you are told that the graph of $y = f(x)$ is vertically stretched by factor $4$, you should write $y = 4f(x)$. If the graph is horizontally stretched by factor $3$, you should write $y = f\left(\frac{x}{3}\right)$.

If a question gives a transformed graph, you may be asked to identify the stretch. For example, if the graph of $y = x^3$ becomes $y = 5x^3$, then every $y$-value has been multiplied by $5$. That is a vertical stretch by factor $5$. If the graph becomes $y = \left(\frac{x}{5}\right)^3$, then it is a horizontal stretch by factor $5$.

Function notation is very important here. You should understand that $f(2x)$ is not a vertical stretch by factor $2$. Instead, it is a horizontal compression by factor $\frac{1}{2}$. This is because the factor is inside the function, attached to $x$.

Examples with common functions

Linear functions

For $f(x) = x$, a vertical stretch gives $y = 4x$. This graph passes through the origin and has a larger gradient. A horizontal stretch does not change the fact that the graph is still a straight line through the origin, but the formula becomes $y = \frac{x}{4}$ if you write it as $f\left(\frac{x}{4}\right)$.

Quadratic functions

For $f(x) = x^2$, the graph of $y = 3x^2$ is vertically stretched and narrower. The graph of $y = \left(\frac{x}{3}\right)^2$ is horizontally stretched and wider. The vertex stays at $(0,0)$ because stretches do not move the graph.

Exponential functions

For $f(x) = 2^x$, the graph of $y = 2\cdot 2^x$ is a vertical stretch by factor $2$. The graph of $y = 2^{x/2}$ is a horizontal stretch by factor $2$. Both graphs show growth, but the rate at which the values increase appears different.

Logarithmic functions

For $f(x) = \log(x)$, a vertical stretch gives $y = 3\log(x)$. This makes the graph steeper. A horizontal stretch gives $y = \log\left(\frac{x}{3}\right)$. The graph spreads out more along the $x$-axis.

Rational functions

For $f(x) = \frac{1}{x}$, a vertical stretch by factor $2$ gives $y = \frac{2}{x}$. The branches move farther from the axes. A horizontal stretch by factor $2$ gives $y = \frac{1}{x/2} = \frac{2}{x}$, which is interesting because it gives the same algebraic expression. This shows that some functions can simplify in ways that make different transformations look identical after algebra. For this reason, it is important to think about both the formula and the graph.

Stretches and key features of graphs

Stretches affect the size of a graph, but not every feature changes in the same way.

A vertical stretch changes $y$-coordinates but leaves $x$-coordinates the same.
A horizontal stretch changes $x$-coordinates but leaves $y$-coordinates the same.
Intercepts may change, depending on the function.
Turning points, roots, asymptotes, and domain/range can also be affected.

For example, if $f(x)$ has a root at $x = 3$, then the vertically stretched graph $y = 2f(x)$ still has the root at $x = 3$, because the $x$-value where $f(x) = 0$ does not change. But the graph may rise or fall more steeply around that point.

For a horizontal stretch, roots move. If $f(3) = 0$, then $y = f\left(\frac{x}{2}\right)$ has a root where $\frac{x}{2} = 3$, so $x = 6$. The root moves farther from the $y$-axis.

This idea is especially useful in exam questions that ask you to sketch transformed graphs quickly and accurately.

Conclusion

Stretches of graphs are a key part of function transformations in IB Mathematics Analysis and Approaches SL. They help you understand how changing a function’s rule changes its graph. A vertical stretch uses $y = af(x)$ and changes outputs, while a horizontal stretch uses $y = f\left(\frac{x}{a}\right)$ and changes inputs. students, if you can recognize these forms, interpret them on graphs, and connect them to real-world models, you will be much stronger in the entire Functions topic. Stretches are not just a graphing trick—they are a way to describe how patterns change in mathematics and in the real world 🌟

Study Notes

A stretch changes the size of a graph without changing its basic shape.
A vertical stretch by factor $a$ is written as $y = af(x)$.
If $a > 1$, the graph is stretched away from the $x$-axis.
If $0 < a < 1$, the graph is compressed toward the $x$-axis.
A horizontal stretch by factor $a$ is written as $y = f\left(\frac{x}{a}\right)$.
If $a > 1$, the graph is stretched away from the $y$-axis.
If $0 < a < 1$, the graph is compressed toward the $y$-axis.
A negative vertical factor, such as $y = -af(x)$, includes a reflection in the $x$-axis.
Horizontal stretches change the graph in the opposite way inside the function.
Stretches are important for linear, quadratic, rational, exponential, and logarithmic functions.
Vertical stretches change $y$-coordinates; horizontal stretches change $x$-coordinates.
In exam questions, always check whether the factor is outside or inside the function.