Vertical Stretch: Exponential Function Explained

Hey guys! Ever wondered what makes an exponential graph shoot up or flatten out? Well, today we're diving deep into the world of vertical stretches and how they specifically mess with exponential functions. You know, those functions that grow or shrink super fast? We're going to break down exactly which part of the equation is the secret sauce for stretching things vertically, and we'll use some examples to make it crystal clear. So, grab your notebooks, or just kick back and let's unravel this math mystery together! We'll be looking at a specific question: Which function represents a vertical stretch of an exponential function? And we'll explore the options: A. $f(x)=3 ext{ (1/2)}^x$, B. $f(x)=1/2 ext{ (3)}^x$, C. $f(x)=(3)^{2x}$, D. $f(x)=3^{(1/2)x}$. Understanding these transformations is key to really grasping how functions behave, and trust me, once you get it, it's like unlocking a new superpower in your math toolkit.

Understanding Exponential Functions and Transformations

Alright, let's get down to brass tacks. Before we can talk about stretching, we need to get a solid handle on what an exponential function actually is and what a vertical stretch does. An exponential function, in its most basic form, looks something like $f(x) = a imes b^x$. Here, '$b$' is the base, and it's what determines the rate of growth or decay. If '$b$' is greater than 1, the function grows; if '$b$' is between 0 and 1, it decays. The '$a$' part? That's our multiplier. Now, what's a vertical stretch? Imagine you have a graph, and you grab the top and bottom of it and pull it upwards, making it taller. That's a vertical stretch! In mathematical terms, a vertical stretch happens when you multiply the entire function by a constant factor greater than 1. So, if you have a function $g(x)$, a vertical stretch would result in a new function, let's call it $h(x)$, where $h(x) = k imes g(x)$, and '$k$' is that stretch factor, with $k > 1$. If '$k$' is between 0 and 1, it's actually a vertical compression, making the graph shorter. So, the key takeaway here is that the stretching or compressing happens after the base function has been evaluated. It’s like you’re taking the output of the original function and then scaling it up or down. This is super important because it differentiates a vertical stretch from other transformations, like horizontal stretches or shifts.

Identifying the Vertical Stretch in Exponential Functions

Now, let's put our knowledge of vertical stretches to work on exponential functions. We're looking for a function that represents a vertical stretch. Remember, a vertical stretch means we're multiplying the entire original exponential function by a constant factor. Let's consider the general form of an exponential function, $f(x) = a imes b^x$. If we want to apply a vertical stretch by a factor of 'k', the new function will be $g(x) = k imes (a imes b^x)$. See how the '$k$' is multiplying the whole thing? Now, let's look at our options and see which one fits this pattern.

• Option A: $f(x)=3 ext{ (1/2)}^x$ In this case, the base is $1/2$, and the multiplier in front is 3. This looks like our '$a$' value is 3 and our '$b$' value is $1/2$. This is a vertical stretch because the multiplier (3) is greater than 1. It's stretching the basic exponential function $g(x) = (1/2)^x$ by a factor of 3. So, this one is a strong contender!

• Option B: $f(x)=1/2 ext{ (3)}^x$ Here, the base is 3, and the multiplier is $1/2$. This means we're taking the exponential function $g(x) = 3^x$ and multiplying it by $1/2$. Since $1/2$ is less than 1, this represents a vertical compression, not a stretch. So, this one is out.

• Option C: $f(x)=(3)^{2x}$ This one looks a bit different. We can rewrite this using exponent rules as $f(x) = (3^2)^x = 9^x$. Here, the base is effectively 9, and there's no separate multiplier in front of the entire expression. This represents a different kind of change to the growth rate, essentially making the base larger, which leads to faster growth, but it's not a vertical stretch of a basic exponential function like $3^x$.

• Option D: $f(x)=3^{(1/2)x}$ Similar to option C, we can rewrite this as $f(x) = (3^{1/2})^x = ( ext{sqrt}(3))^x$. Again, this changes the base of the exponential function, making it smaller (since $ ext{sqrt}(3)$ is about 1.732), which slows down the growth. It's not a vertical stretch.

So, based on our analysis, Option A: $f(x)=3 ext{ (1/2)}^x$ is the one that clearly represents a vertical stretch of an exponential function. The factor of 3 is multiplying the entire expression $(1/2)^x$, making the graph of $(1/2)^x$ taller.

How Vertical Stretches Affect the Graph

Let's really dig into why Option A represents a vertical stretch and what that actually looks like on a graph, guys. We’re talking about taking a function, say $g(x) = (1/2)^x$, and then applying a vertical stretch by a factor of 3 to get $f(x) = 3 imes (1/2)^x$. What does this do? For every single '$x$' value you input, the output of $f(x)$ will be three times the output of $g(x)$. Let's take a few points.

Consider $g(x) = (1/2)^x$:

• If $x = 0$, $g(0) = (1/2)^0 = 1$. The point is $(0, 1)$.
• If $x = 1$, $g(1) = (1/2)^1 = 1/2$. The point is $(1, 1/2)$.
• If $x = 2$, $g(2) = (1/2)^2 = 1/4$. The point is $(2, 1/4)$.

Now, let's look at $f(x) = 3 imes (1/2)^x$:

• If $x = 0$, $f(0) = 3 imes (1/2)^0 = 3 imes 1 = 3$. The point is $(0, 3)$.

• If $x = 1$, $f(1) = 3 imes (1/2)^1 = 3 imes 1/2 = 3/2$. The point is $(1, 3/2)$.

• If $x = 2$, $f(2) = 3 imes (1/2)^2 = 3 imes 1/4 = 3/4$. The point is $(2, 3/4)$.

See what happened? The y-values have all been multiplied by 3. The point $(0, 1)$ on $g(x)$ becomes $(0, 3)$ on $f(x)$. The point $(1, 1/2)$ becomes $(1, 3/2)$. The graph has been stretched upwards, making it appear