Hasil Dilatasi Horizontal Fungsi Kuadrat

Hey guys, let's dive into the awesome world of math and tackle a cool problem about function transformations! Today, we're going to figure out the equation of a function after it's been horizontally dilated. Specifically, we're working with the function $y = 3x^2 - 9x - 27$ and we're going to dilate it horizontally with a scale factor of 3. This means we're stretching or squishing the graph of the function along the x-axis. It's a super common topic in algebra and pre-calculus, and understanding it will seriously level up your graphing game. So, grab your notebooks, maybe a snack, and let's get this done!

Understanding Horizontal Dilations

Alright, first things first, what exactly is a horizontal dilation? When we talk about dilating a function horizontally, we're essentially changing how wide or narrow its graph appears. A horizontal dilation with a scale factor of $k$ affects the input values (the $x$-values) of the function. If you have a function $f(x)$, a horizontal dilation by a factor of $k$ results in a new function $g(x) = f(x/k)$. Notice that we replace $x$ with $x/k$. This might seem a little counter-intuitive at first – you might expect to replace $x$ with $kx$. But think about it this way: to get the same $y$-value as the original function $f(x)$ at a certain point, the new function $g(x)$ needs to be evaluated at a different $x$-value. If $k > 1$, the graph is stretched horizontally, meaning you need a larger $x$-value in $g(x)$ to get the same output as $f(x)$ at a smaller $x$-value. If $0 < k < 1$, the graph is compressed horizontally.

In our specific problem, the original function is $f(x) = 3x^2 - 9x - 27$, and the horizontal scale factor is $k=3$. So, to find the new function, which we can call $g(x)$, we need to substitute $x/3$ for every $x$ in the original function $f(x)$. This is the core concept we need to apply. Don't get confused with vertical dilations, where you multiply the entire function by the scale factor ($k imes f(x)$). Horizontal dilations mess with the 'inside' of the function, affecting the $x$-variable directly. It's like looking at something through a magnifying glass that only stretches things left and right. Pretty neat, huh? So, keep this rule in mind: for a horizontal dilation by a factor of $k$, replace $x$ with $x/k$. Let's go!

Applying the Transformation

Now that we've got the concept down, let's get our hands dirty with the actual calculation. Our original function is $f(x) = 3x^2 - 9x - 27$. We need to perform a horizontal dilation with a scale factor of $k=3$. As we established, this means we replace every instance of $x$ in the function's formula with $x/3$. So, our new function, let's call it $g(x)$, will be $g(x) = f(x/3)$.

Let's substitute $(x/3)$ into our function step-by-step:

1. The $x^2$ term: The original term is $3x^2$. Replacing $x$ with $x/3$ gives us $3(x/3)^2$. Let's simplify this: $3(x^2/9) = 3x^2/9 = x^2/3$.
2. The $x$ term: The original term is $-9x$. Replacing $x$ with $x/3$ gives us $-9(x/3)$. Simplifying this yields $-9x/3 = -3x$.
3. The constant term: The constant term is $-27$. Since there's no $x$ here, it remains unchanged.

Putting it all together, the new function $g(x)$ is the sum of these transformed terms: $g(x) = (x^2/3) - 3x - 27$. So, the equation of the function after the horizontal dilation is $y = \frac{1}{3}x^2 - 3x - 27$. This is our final answer for the transformed function. It looks a bit different from the original, and that's exactly what we expect when we apply transformations. The shape is still parabolic, but it's been stretched out horizontally.

Let's recap the process: identify the original function $f(x)$, identify the horizontal scale factor $k$, and then substitute $x/k$ for every $x$ in $f(x)$. Then, simplify the resulting expression. Easy peasy, right? This method works for any function, not just quadratic ones. Keep this in mind as you encounter more transformation problems.

Comparing with Options (if applicable)

Sometimes, when you're working on problems, you might be given multiple-choice options. If this were a multiple-choice question, our derived function $y = \frac{1}{3}x^2 - 3x - 27$ would be the correct choice. Let's briefly look at why the other options might be incorrect to solidify our understanding. The options provided were:

• $y = x^2 - 3x - 9$
• $y = 9x^2 - 27x - 81$
• $y = x^2 - 3x - 27$

Our calculated result is $y = \frac{1}{3}x^2 - 3x - 27$. None of the options exactly match our result. This is a crucial point, guys. It's possible there was a typo in the provided options, or perhaps the question intended a different transformation. However, based strictly on the problem statement – a horizontal dilation of $y = 3x^2 - 9x - 27$ with a factor of 3 – our derived equation $y = \frac{1}{3}x^2 - 3x - 27$ is the mathematically correct outcome.

Let's analyze the other options to see what transformations they might represent:

• Option 1: $y = x^2 - 3x - 9$ This looks like it involves changes to the coefficients and the constant term. If we were to compare it to the original $f(x) = 3x^2 - 9x - 27$, notice that if we first divide the original function by 3, we get $f(x)/3 = (3x^2 - 9x - 27)/3 = x^2 - 3x - 9$. This would represent a vertical compression by a factor of 3, not a horizontal dilation.

• Option 2: $y = 9x^2 - 27x - 81$ This looks like $3 imes (3x^2 - 9x - 27)$. So, $y = 3 imes f(x)$. This represents a vertical stretch by a factor of 3.

• Option 3: $y = x^2 - 3x - 27$ This option is close to our answer but has a different coefficient for the $x^2$ term. It seems like a mix-up. If the original function was $y = x^2 - 9x - 27$, and we performed a horizontal dilation by 3, we would replace $x$ with $x/3$: $y = (x/3)^2 - 9(x/3) - 27 = x^2/9 - 3x - 27$. Still not this option.

So, to reiterate, based on the strict definition of a horizontal dilation by a factor of 3 on the function $y = 3x^2 - 9x - 27$, the correct equation is $y = \frac{1}{3}x^2 - 3x - 27$. If you encountered this problem in a test and got this result, and none of the options matched, it's important to double-check your work and then perhaps note the discrepancy. It's also possible the question meant a vertical dilation or a different scale factor, but we must answer the question as written.

Conclusion and Key Takeaways

Alright guys, we've successfully navigated the process of performing a horizontal dilation on a quadratic function. Remember, the key to horizontal transformations is that they affect the input variable, $x$. For a horizontal dilation by a scale factor $k$, you replace every $x$ in the function with $x/k$. For our specific problem, the original function was $f(x) = 3x^2 - 9x - 27$, and we applied a horizontal dilation with $k=3$. This led us to substitute $x/3$ for $x$, resulting in the new function $g(x) = 3(x/3)^2 - 9(x/3) - 27$.

After simplifying, we found that $g(x) = 3(x^2/9) - 3x - 27$, which further simplifies to $g(x) = x^2/3 - 3x - 27$. So, the equation after the horizontal dilation is $y = \frac{1}{3}x^2 - 3x - 27$. This is the precise mathematical outcome of the transformation described. It's crucial to distinguish this from vertical dilations, where you would multiply the entire function by the scale factor. A horizontal dilation stretches or compresses the graph along the x-axis, changing its width, while a vertical dilation stretches or compresses it along the y-axis, changing its height.

Here are the main takeaways from this session:

1. Horizontal Dilation Rule: To dilate a function $f(x)$ horizontally by a factor of $k$, replace $x$ with $x/k$ in the function's expression. The new function is $g(x) = f(x/k)$.
2. Applying to Quadratics: For $f(x) = ax^2 + bx + c$, a horizontal dilation by $k$ yields $g(x) = a(x/k)^2 + b(x/k) + c$.
3. Simplification is Key: Always simplify the resulting expression to get the final equation of the transformed function.
4. Distinguish from Vertical Dilations: Remember that a vertical dilation by $k$ results in $k imes f(x)$, which is different from a horizontal dilation.

Understanding these transformations is fundamental in mastering function analysis and graphing. Practice with different functions and scale factors to build your confidence. Keep experimenting, keep learning, and you'll become a math whiz in no time! Happy graphing!