Vertical Stretch

Vertical stretch is a function transformation in Honors Pre-Calculus where every y-value is multiplied by a constant, making the graph taller or shorter. If the constant is greater than 1, the graph stretches; if it is between 0 and 1, it compresses.

What is Vertical Stretch?

Vertical stretch is what happens when you multiply a function’s output by a constant in Honors Pre-Calculus. If you start with $f(x)$ and write $g(x)=a f(x)$, every point on the graph moves vertically away from or toward the x-axis by the factor $a$.

When $a>1$, the graph gets taller. The y-values increase in size, so peaks rise higher and valleys drop farther down. When $0<a<1$, the graph gets compressed vertically, which makes the graph look flatter. The x-values do not change, so the graph keeps the same left-to-right placement.

A good way to picture this is to compare a parent function to its transformed version. If $f(x)=x^2$ and $g(x)=3x^2$, then every output is tripled. The vertex stays in the same place, but the parabola opens with steeper sides. If $g(x)=\tfrac12 x^2$, the parabola is wider-looking because all the y-values are cut in half.

This transformation shows up in several parts of the course. For sine and cosine, the coefficient in front controls amplitude, which is really a vertical stretch or compression. For absolute value, the same coefficient changes how steep the V-shape looks. For power, polynomial, and logarithmic functions, the vertical factor changes the graph’s height without changing the basic parent-function structure.

One common mistake is mixing up vertical stretch with horizontal stretch. Vertical stretch changes output values, so you multiply the whole function by a constant. Horizontal stretch changes input values, so the constant goes inside the function. If you keep asking, “Did the y-values change or did the x-values change?” you can tell them apart fast.

Why Vertical Stretch matters in Honors Pre-Calculus

Vertical stretch shows up everywhere in Honors Pre-Calculus because it is one of the fastest ways to change a graph without rebuilding it from scratch. Once you know the parent function, the coefficient in front tells you how the graph has been scaled, which makes sketching much quicker and more accurate.

This is especially useful with sine and cosine. The coefficient becomes amplitude, so you can read how tall the wave is and model situations like sound waves, tides, or other repeating patterns. A bigger coefficient means bigger oscillations, while a fraction makes the wave flatter.

You also see vertical stretch in absolute value graphs, where it changes the steepness of the V, and in power or polynomial functions, where it can make a curve rise more sharply or flatten out. Even with logarithmic graphs, multiplying the function changes the vertical scale, which affects how the graph is interpreted on a coordinate plane.

In problem solving, this lets you match equations to graphs, write equations from a graph, and explain transformations clearly. If the graph looks taller but stays centered in the same place, vertical stretch is usually the first transformation to check.

How Vertical Stretch connects across the course

Transformation of Functions

Vertical stretch is one type of function transformation. In a full transformation problem, you may see shifts, reflections, and stretches together, so you need to separate what changes the y-values from what changes the x-values. Vertical stretch is the output-scaling piece of that bigger toolkit.

Amplitude

For sine and cosine graphs, vertical stretch directly controls amplitude. If the function is $y=a\sin x$ or $y=a\cos x$, the absolute value of $a$ tells you how far the wave rises above and falls below the midline. That is why amplitude is basically the vertical stretch of a trig graph.

Parent Function

You usually identify vertical stretch by starting with the parent function. The parent gives you the original shape, like $x^2$, $|x|$, or $\sin x$. Then you compare the transformed graph to that original shape and ask how the y-values changed.

Absolute Value

Absolute value graphs make vertical stretch easy to see because the V-shape gets steeper or flatter. If the coefficient in front of $|x|$ is greater than 1, the arms of the V rise faster. If it is between 0 and 1, the graph opens more loosely.

Is Vertical Stretch on the Honors Pre-Calculus exam?

A graphing question usually asks you to identify how a function changed from its parent, or to sketch the new graph from an equation like $g(x)=2f(x)$. The move is simple: check the coefficient outside the function and decide whether the graph is stretched or compressed vertically. For trig graphs, use that coefficient to find amplitude. For absolute value, polynomial, or logarithmic graphs, describe how the y-values changed while the x-values stayed in place. If you are given a graph and asked for an equation, compare the height of key points to the parent function and solve for the scale factor. A common mistake is treating the stretch like a shift, so always look for multiplication, not addition.

Vertical Stretch vs Horizontal Stretch

Vertical stretch changes the output values, so it affects height, amplitude, or steepness. Horizontal stretch changes the input values, so it affects width or how quickly the graph spreads left and right. A quick check: if the constant is outside the function, it is vertical; if it is inside, it is horizontal.

Key things to remember about Vertical Stretch

• Vertical stretch multiplies the outputs of a function by a constant, so it changes the graph’s height without changing its basic shape.

• If the constant is greater than 1, the graph stretches vertically; if it is between 0 and 1, the graph compresses vertically.

• For sine and cosine, vertical stretch is the same idea as amplitude.

• The x-values stay in the same places, so vertical stretch is not the same thing as a horizontal stretch.

• A fast way to spot it is to look for a number outside the function, like $a f(x)$.

Frequently asked questions about Vertical Stretch

What is vertical stretch in Honors Pre-Calculus?

Vertical stretch is a transformation that multiplies all of a function’s y-values by a constant. If the constant is bigger than 1, the graph gets taller; if it is between 0 and 1, the graph gets shorter or flatter. The overall shape stays the same.

How do you know if a function is vertically stretched or compressed?

Look at the coefficient outside the function. If $a>1$, the graph is vertically stretched, and if $0<a<1$, it is vertically compressed. The easiest clue is that the change happens to y-values, not x-values.

What is the difference between vertical stretch and horizontal stretch?

Vertical stretch changes the outputs of the function, so it changes height or amplitude. Horizontal stretch changes the inputs, so it changes width and the spacing of points. If the number is outside the function, think vertical; if it is inside, think horizontal.

How does vertical stretch affect a sine or cosine graph?

It changes the amplitude. A larger coefficient makes the wave taller, and a smaller positive coefficient makes it flatter. The period does not change just because of a vertical stretch.