Understanding Vertical Compression in Functions

Which statement is true when k is a fraction in the expression k×f(x)?

• A. It leads to a vertical stretch
• B. It results in horizontal compression
• C. It causes a vertical compression
• D. It reflects the function over the y-axis

Answer: (C)

In the context of the function \( f(x) \) when multiplied by a fraction \( k \), the resulting transformation affects the graph of the function in a specific manner. When \( k \) is a fraction, it represents a number less than 1 and greater than 0. This applies a vertical compression to the function. Specifically, a vertical compression means that every point on the graph of the function is pulled closer to the x-axis. For example, if \( f(x) \) has a value of 2 at a certain point, then \( k \times f(x) \) will have a value that is a fraction of that, effectively making the output smaller. This is why the correct statement involves vertical compression: the effect of multiplying by a fraction reduces the height of the graph without changing its overall direction. In contrast, a vertical stretch would occur if \( k \) were greater than 1, resulting in points moving away from the x-axis, while horizontal compression would relate to multiplying \( x \) by a fraction less than 1 inside the function. A reflection over the y-axis does not apply here since that transformation occurs when \( x \) is replaced with \(-x\) in the function.

When studying for the Ohio Assessments for Educators (OAE) Mathematics exam, understanding function transformations becomes vital. One crucial aspect to grasp is what happens when you apply a fraction to a function. Have you ever wondered how multiplying a function by a fraction affects its graph? Well, let’s dive into that!

Imagine you have a function ( f(x) ). Now, if you multiply this function by a fraction ( k ) (where ( k ) is, say, 0.5 or 0.25), what do you think will happen? The truth is, this leads to a vertical compression of the function’s graph. Sounds a bit complex? It's not!

When ( k ) is a fraction, it represents a number less than 1 but greater than 0. So, let’s say at a specific point, ( f(x) ) gives out a value of 2. If you multiply this by ( k ) (for instance, 0.5), you will get 1. This means every point on the graph is being pulled closer to the x-axis. Such a transformation shrinks the height of the function's graph without changing its overall direction. If ( f(x) ) were to represent a mountain, applying that fraction turns that mountain into a hill—less steep, but still in the same spot!

Now, what’s the opposite of vertical compression? Right! A vertical stretch. This happens when ( k ) is greater than 1. Imagine if that mountain grew taller instead, pushing its peak further into the sky. This can be a handy mental image when considering how these transformations affect your function.

You might also be wondering about horizontal compression. This one’s a little different—it occurs when you multiply ( x ) by a fraction less than 1 within the function itself. So think of it as squeezing the graph from the sides—it gets narrower and taller, while vertical compression makes it look flatter.

And here’s where it gets interesting! The idea of reflections comes into play too. When we talk about reflecting a graph, that happens when you swap ( x ) for (-x). So, that’s a whole separate transformation not related to the fraction in front of ( f(x) ).

Feeling more comfortable with this, yet? Understanding these transformations not only prepares you for the OAE but helps in teaching others too! I mean, how cool is it to be able to show a student how a simple fraction can totally change the way a function looks on a graph? It's like magic—math magic!

So, as you prepare for your math assessments, keep these key concepts in mind. Knowing how vertical compression works will give you a solid edge, helping you tackle any related questions that might come your way. Trust me; mastering these fundamentals will go a long way in your success in the examination. And as always, remember to embrace the journey of learning—each concept adds another layer to your understanding!