Vertical stretch
Vertical stretch is a function transformation that multiplies every y-value by a number greater than 1, making the graph taller. In College Algebra, you use it to change amplitude or range without changing the basic shape.
What is vertical stretch?
Vertical stretch in College Algebra means you multiply the output of a function by a constant, usually written as y = af(x). When a is greater than 1, every point on the graph moves farther from the x-axis, so the graph looks taller.
The easiest way to picture it is to think about the y-values only. If a point on the parent function is (x, y), the stretched version becomes (x, ay). The x-values stay the same, which is why the graph keeps the same basic shape. Only the vertical distance changes.
A vertical stretch is different from a shift. A shift moves the graph up, down, left, or right. A stretch changes how steep, tall, or spread out the graph looks vertically. That matters a lot when you are graphing families of functions, because you can start with a parent function and adjust it without redrawing everything from scratch.
This shows up a lot in exponential and trigonometric graphs. For a sine or cosine graph, the coefficient in front controls amplitude, so y = 3sin(x) has three times the amplitude of y = sin(x). For an exponential function like y = 2(3^x), the 2 is a vertical stretch that doubles every output of the parent function 3^x. The graph still grows the same way horizontally, but the y-values are scaled up.
One common mistake is treating vertical stretch like horizontal stretch. They are not the same. A vertical stretch changes outputs after the function is evaluated, while a horizontal stretch changes the input and affects the graph in the opposite direction. If you are checking your work, look at whether the factor is outside the function or inside the parentheses. Outside usually means vertical stretch.
Why vertical stretch matters in College Algebra
Vertical stretch shows up any time you need to read or build a transformed graph from an equation. In College Algebra, that means more than just sketching pictures. You may be asked to identify how a graph changed from its parent function, compare two functions, or explain why one graph has a larger amplitude or range than another.
It is especially useful in graphing exponential functions and trigonometric functions. For exponentials, a vertical stretch changes the starting height and every later output, which can make growth or decay curves look steeper on a graph. For sine and cosine, the stretch changes amplitude, which is how you describe the height of waves in problems about sound, tides, or any repeating pattern.
It also connects to algebraic thinking. When you see a coefficient in front of a function, you need to know what it does instead of guessing. That skill saves time on graphing problems and helps you write equations from given graphs. If a graph is taller than the parent but still has the same basic shape, vertical stretch is usually the move you are looking for.
Keep studying College Algebra Unit 6
Visual cheatsheet
view galleryHow vertical stretch connects across the course
Vertical Compression
Vertical compression is the flip side of vertical stretch. Instead of multiplying outputs by a number greater than 1, you multiply by a number between 0 and 1, which pulls the graph closer to the x-axis. In practice, both transformations keep the same x-values and the same general shape, but they change how tall the graph looks.
Amplitude
Amplitude is the part of a sine or cosine graph that vertical stretch changes. If you multiply the function by 4, the amplitude becomes four times as large. That is why the coefficient in front of trig functions matters so much when you are reading or writing equations from graphs.
Parent Function
A parent function is the simplest version of a graph before any transformations. Vertical stretch is easiest to spot when you compare a graph to its parent, because you can see that the shape stays the same while the y-values get scaled. Starting from the parent function is the usual graphing strategy in College Algebra.
Horizontal Stretch
Horizontal stretch can look similar to vertical stretch at first, but it changes the graph in a different direction. A horizontal stretch affects x-values, which changes how wide the graph looks. Vertical stretch affects y-values, so the graph gets taller instead of wider. Knowing which variable gets multiplied helps you avoid mixing them up.
Is vertical stretch on the College Algebra exam?
A graphing quiz or problem set item may give you a function like y = 2f(x) and ask you to describe the transformation or sketch the new graph. Your job is to recognize that every output is doubled, so the graph is vertically stretched by a factor of 2. If the problem gives a sine or cosine graph, you may need to read the amplitude from the coefficient or compare it to the parent function. For exponential graphs, you might explain how the stretch changes the y-intercept and overall height without changing the base growth pattern. The fastest move is to check whether the number is outside the function, since that signals a vertical change.
Vertical stretch vs Horizontal Stretch
Horizontal stretch is commonly confused with vertical stretch because both make graphs look larger. The difference is where the change happens. Vertical stretch multiplies the outputs, so the graph gets taller. Horizontal stretch changes the inputs, so the graph gets wider. If the factor is outside the function, think vertical. If it is inside the input, think horizontal.
Key things to remember about vertical stretch
Vertical stretch multiplies a function’s y-values, so the graph becomes taller without changing its basic shape.
A factor greater than 1 creates a stretch, while a factor between 0 and 1 creates a vertical compression.
In trigonometric functions, vertical stretch changes amplitude but does not change period or frequency.
In exponential functions, vertical stretch scales the outputs and changes the graph’s height, including the y-intercept.
If the number is outside the function, you are usually looking at a vertical transformation.
Frequently asked questions about vertical stretch
What is vertical stretch in College Algebra?
Vertical stretch is a transformation that multiplies a function’s outputs by a constant greater than 1. The graph gets taller, but its basic shape stays the same. In College Algebra, you see it when graphing transformed parent functions, especially exponentials and trig graphs.
How do you know if a function is vertically stretched?
Look for a coefficient outside the function, like y = 3f(x). That means every y-value is multiplied by 3, so the graph is vertically stretched. If the coefficient is between 0 and 1, the graph is vertically compressed instead.
Does vertical stretch change the period of a graph?
No. Vertical stretch changes the height of the graph, not how quickly it repeats. For sine and cosine, the period stays the same because the x-values are unchanged. The same idea applies to other function families where the stretch is only outside the function.
What is the difference between vertical stretch and amplitude?
Vertical stretch is the transformation, and amplitude is one feature it changes in sine and cosine graphs. If you multiply a trig function by a number, the amplitude changes by that factor. So amplitude is the result you see, while vertical stretch is the rule causing it.