Finding The Range Of Quadratic Function F(x)=(x-3)^2-7

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Hey guys! Today, we're diving into the fascinating world of quadratic functions. Specifically, we're going to figure out how to find the range of a quadratic function. We'll use the example f(x) = (x - 3)² - 7. Don't worry, it's not as scary as it looks! We'll break it down step-by-step so you can confidently tackle these problems. Understanding the range is super important because it tells us all the possible output values (or y-values) our function can produce. So, let's get started and unravel this mathematical mystery together!

Understanding Quadratic Functions

Before we jump into finding the range, let's quickly recap what a quadratic function actually is. In its most basic form, a quadratic function looks like this: f(x) = ax² + bx + c, where a, b, and c are constants (just fancy numbers, really!) and a can't be zero (otherwise, it wouldn't be quadratic anymore!). Our function, f(x) = (x - 3)² - 7, is also a quadratic function, but it's written in a slightly different form called vertex form. We'll see why that's helpful in a bit.

The graph of a quadratic function is a parabola, which is a U-shaped curve. This shape is crucial to understanding the range. Parabolas can open upwards or downwards, depending on the value of a. If a is positive, the parabola opens upwards, forming a smiley face. If a is negative, it opens downwards, like a frowny face. The vertex of the parabola is the point where the curve changes direction – it's either the lowest point (if the parabola opens upwards) or the highest point (if it opens downwards). This vertex plays a key role in determining the range.

Key takeaway: Quadratic functions form parabolas, and the direction and vertex of the parabola are essential for finding the range. Now that we have a solid grasp of the basics, let's move on to how we can leverage this information to solve our problem.

Identifying the Vertex

The vertex is the cornerstone for determining the range of our quadratic function. Our given function, f(x) = (x - 3)² - 7, is conveniently presented in vertex form: f(x) = a(x - h)² + k. In this form, the vertex of the parabola is simply the point (h, k). This is a huge advantage because we can directly read off the vertex from the equation. In our case, h = 3 and k = -7. So, the vertex of our parabola is (3, -7). Remember, the vertex is the point where the parabola changes direction, making it either the minimum or maximum point of the function.

But what does the vertex tell us about the range? Well, it depends on whether the parabola opens upwards or downwards. To figure that out, we look at the coefficient a in the vertex form. In our function, a is implicitly 1 (since there's no number explicitly multiplying the squared term). Because 1 is positive, our parabola opens upwards. This means the vertex is the minimum point of the function. The y-coordinate of the vertex, which is -7, is the lowest possible y-value our function can have. All other y-values will be greater than -7. Think of it like a valley – the vertex is the bottom of the valley, and the parabola extends upwards from there.

Key takeaway: By identifying the vertex (3, -7) and noting that the parabola opens upwards, we know the minimum y-value of our function is -7. This is a critical piece of the puzzle for finding the range. Now that we've pinpointed the vertex, we're well on our way to defining the range.

Determining the Direction of the Parabola

The direction in which our parabola opens is paramount in determining the function's range. We already touched on this when identifying the vertex, but let's delve a bit deeper. As a reminder, the coefficient a in the vertex form f(x) = a(x - h)² + k dictates the parabola's direction. If a is positive, the parabola opens upwards; if a is negative, it opens downwards.

In our function, f(x) = (x - 3)² - 7, the coefficient a is 1, which is undeniably positive. Therefore, our parabola opens upwards. This is excellent news because it tells us that our vertex, (3, -7), represents the minimum point on the graph. This means the y-value of the vertex, -7, is the smallest possible output of our function. All other points on the parabola will have y-values greater than -7. Imagine pouring water into the parabola – it would fill up from the bottom (the vertex) upwards, never going below the y-value of -7.

If our parabola opened downwards (if a were negative), the vertex would be the maximum point, and the range would consist of all y-values less than or equal to the y-value of the vertex. But since our parabola opens upwards, we know we're dealing with a minimum y-value. Key takeaway: The positive coefficient a confirms that our parabola opens upwards, making the vertex the minimum point and setting the stage for defining the range. Understanding the direction is half the battle, and we've conquered it!

Defining the Range

Now for the grand finale: defining the range! We've done all the groundwork, so this part is a breeze. Remember, the range is the set of all possible output values (or y-values) of our function. We've established that our parabola opens upwards and that the vertex (3, -7) is the minimum point.

This means that the smallest possible y-value our function can produce is -7. And since the parabola opens upwards, it extends infinitely upwards, meaning there's no upper limit to the y-values. Therefore, the range includes -7 and all values greater than -7. We can express this range in a couple of ways. One way is using inequality notation: y ≥ -7. This simply means that y is greater than or equal to -7.

Another way to express the range is using interval notation: [-7, ∞). This notation indicates that the range starts at -7 (inclusive, thanks to the square bracket) and extends to positive infinity (indicated by the parenthesis, since infinity isn't a specific number we can include). Key takeaway: We've successfully defined the range of f(x) = (x - 3)² - 7 as y ≥ -7 or [-7, ∞), meaning the function can produce any y-value greater than or equal to -7.

Visualizing the Range

To really solidify our understanding, let's visualize the range on a graph. If you were to plot the function f(x) = (x - 3)² - 7, you'd see a U-shaped parabola with its vertex at (3, -7). The curve extends upwards from this point, never going below y = -7. Imagine drawing a horizontal line at y = -7. The entire parabola sits on or above that line. This visually confirms that the range includes all y-values from -7 upwards.

You can also picture a coordinate plane and imagine shading in all the y-values that are part of the range. You'd start by shading a solid line at y = -7 (because -7 is included in the range) and then shade everything above that line. This shaded region represents all the possible y-values that our function can produce. Key takeaway: Visualizing the graph reinforces that the range is all y-values greater than or equal to -7, stemming from the parabola's upward opening and vertex at (3, -7). A visual representation provides an intuitive check on our algebraic solution.

Conclusion

So, there you have it! We've successfully navigated the process of finding the range of the quadratic function f(x) = (x - 3)² - 7. We started by understanding the basic form of quadratic functions and the significance of the parabola. We then identified the vertex, determined the direction of the parabola, and used this information to define the range as y ≥ -7 or [-7, ∞). We even visualized the range on a graph to solidify our understanding.

The key takeaways are:

  • Vertex form f(x) = a(x - h)² + k makes it easy to identify the vertex (h, k).
  • The sign of a determines the direction of the parabola (positive = upwards, negative = downwards).
  • The vertex is the minimum point if the parabola opens upwards and the maximum point if it opens downwards.
  • The range is determined by the y-value of the vertex and the direction of the parabola.

Finding the range of quadratic functions might seem daunting at first, but by breaking it down into these steps, you can confidently tackle any similar problem. Keep practicing, and you'll become a quadratic function whiz in no time! Remember, math is like a puzzle, and we just solved one piece of the bigger picture. Keep exploring, keep learning, and have fun with it! You got this! 🚀