Understanding Vertical Stretching in Functions

What happens to the graph of f(x) when it is multiplied by a whole number k?

• A. Reflected over the x-axis
• B. Vertically compressed
• C. Only horizontally stretched
• D. Vertically stretched

Answer: (D)

When a function \( f(x) \) is multiplied by a whole number \( k \) (where \( k > 1 \)), the graph of the function undergoes a vertical stretch. This transformation effectively means that for every point on the graph of the original function, the y-coordinates are multiplied by \( k \). As a result, if the original function \( f(x) \) has a certain value at a specific \( x \), the new function \( k \cdot f(x) \) will have a larger value at that same \( x \), causing the points of the graph to move further away from the x-axis. For example, if \( k = 2 \), every y-value of the graph is doubled, making the peaks higher and the valleys deeper, thus stretching the graph vertically. The transformation does not affect the shape of the graph horizontally or its overall orientation related to the x-axis. In contrast, a multiplier less than one (but greater than zero) would compress the graph vertically, bringing points closer to the x-axis. Therefore, understanding the effect of multiplying the function by different values is crucial for analyzing how the graph of the function transforms.

Have you ever noticed how a simple mathematical operation can entirely change the way a graph looks? For many students preparing for the Ohio Assessments for Educators (OAE) Mathematics Exam, understanding these transformations is crucial. So, what happens to the graph of a function ( f(x) ) when it's multiplied by a whole number ( k )? You guessed it—let’s dive into the world of vertical stretches!

When you multiply a function by a number greater than 1, say ( k = 2 ), you’re not just playing around with numbers; you're stretching that graph vertically! To simplify it: when every point on the graph has its y-coordinate multiplied by ( k ), the graph is stretched away from the x-axis. Imagine pulling a rubber band; stretching it makes it longer and thinner, right? That's exactly what you're doing with the graph! The peaks rise higher and the valleys dip deeper.

It’s all in the numbers: If the original function at a specific ( x ) had a y-value of 3, the new function ( k \cdot f(x) ) will have a value of ( 2 \cdot 3 = 6 ). So, what does that mean? The entire graph expands, taking it further from the x-axis without changing its horizontal dimensions. Cool, right?

But what if ( k ) were a number between 0 and 1? Well, here’s where it gets interesting—your graph would get compressed! Yup, points would inch closer to the x-axis, like when you squeeze a sponge. This contrast helps you visualize the effect of multiplication on your function.

Why does this matter for the OAE? Well, grasping the impact of these multiplications on function graphs can seriously boost your confidence during the test. Functions are foundational in math, and understanding their transformations can not only help you solve problems but also assist you in teaching these concepts to future students.

When you visualize the graph, think of its peaks and valleys as your mathematical landscape. Each transformation is like rearranging the furniture in your living room—everything looks different but serves the same purpose. Plus, practicing such transformations will get you comfortable with graphing concepts you’ll see in your assessments.

Are you eager to apply this knowledge to your studies? Use this understanding not just for your exams but also in real-life applications like physics or economics, where these function behaviors come into play. The more you connect math to the world around you, the more intuitive it will feel.

So, the next time you're working through mathematics for your OAE preparation, remember: multiplying by whole numbers can stretch your graphs vertically, reshaping the entire mathematical picture. This understanding is a foundational step towards mastering more complex concepts, so take the time to practice and visualize! Happy studying!