Unveiling The Secrets Of A Quadratic Function's Graph

Hey Plastik Magazine readers! Let's dive into the fascinating world of quadratic functions and their graphs. Specifically, we're going to break down the function f(x) = x² - 8x + 5 and figure out which statements about its graph are actually true. This isn't just about math; it's about understanding how these functions behave and how we can visually represent them. So, grab your coffee, get comfy, and let's unravel this mathematical mystery together! We will explore the vertex form, the vertex location, the axis of symmetry, and the y-intercept. These are key features that define the shape and position of a parabola, which is the characteristic curve of a quadratic function. Understanding these elements will allow us to accurately sketch the graph and analyze its properties. Let’s get started and make this journey through the world of parabolas both informative and, dare I say, fun!

Transforming to Vertex Form: The Key to Understanding

Alright, guys, the first statement we're going to investigate is: "The function in vertex form is f(x) = (x - 4)² - 11". To check this, we need to transform our original function, f(x) = x² - 8x + 5, into vertex form. Remember, the vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. We can achieve this transformation by completing the square. Let's do it step by step to make sure we don't miss anything. First, we take the original function: f(x) = x² - 8x + 5. Now, focus on the x² - 8x part. To complete the square, we need to add and subtract the square of half of the coefficient of our x term. The coefficient of our x term is -8. Half of -8 is -4, and (-4)² is 16. So, we'll add and subtract 16. This gives us f(x) = (x² - 8x + 16) - 16 + 5. Notice how we've grouped the first three terms. Those three terms are now a perfect square trinomial, which we can factor into (x - 4)². The remaining terms, -16 and +5, combine to give us -11. Thus, our function in vertex form becomes f(x) = (x - 4)² - 11. This confirms that the first statement is indeed true. Knowing the vertex form is incredibly helpful because it directly reveals the vertex of the parabola, making graphing and analysis much easier. We did it, guys! We successfully transformed the equation and verified our first true statement. Now, we are ready to move on, are you excited?

Locating the Vertex: Where the Magic Happens

Next up, we're looking at the statement: "The vertex of the function is (-8, 5)." This is where things get really interesting, and where understanding the vertex form we just derived comes into play. As a reminder, our function in vertex form is f(x) = (x - 4)² - 11. In vertex form, the vertex is represented by the point (h, k). Looking at our equation, we can see that h is 4 (because it's (x - 4), and the formula is (x - h), so h = 4) and k is -11. Therefore, the vertex of our parabola is (4, -11), not (-8, 5). So, we can definitively say that the second statement is false. The vertex is a super important point because it's the minimum or maximum point of the parabola, depending on whether the parabola opens upwards or downwards. In our case, since the coefficient of the x² term is positive (it's 1), the parabola opens upwards, and the vertex is the minimum point. It’s like the low point on a rollercoaster ride. This helps us visualize the shape and behavior of the graph. It also helps us determine the range of the function, which is the set of all possible y-values. Great job, guys! Now that we know how to identify the vertex correctly, let's keep moving and dig deeper into the problem.

Unveiling the Axis of Symmetry: The Mirror Image

Now, let's check out the statement, "The axis of symmetry is x = 5." The axis of symmetry is a vertical line that passes through the vertex of the parabola, effectively dividing the parabola into two symmetrical halves. Since we know the vertex is at (4, -11), the axis of symmetry is a vertical line that passes through the x-coordinate of the vertex. Therefore, the axis of symmetry is x = 4, not x = 5. Consequently, the third statement is also false. The axis of symmetry is a crucial concept. It tells us where the parabola is perfectly balanced. All points on the parabola are equidistant from this line. It also allows us to quickly find other points on the parabola if we know the coordinates of one point on either side of the axis. Understanding the axis of symmetry is like knowing the central lane on a highway—it guides the whole structure. This understanding not only helps us plot the graph but also provides insights into the behavior and properties of the quadratic function. The axis of symmetry helps determine the symmetry of the graph. It also gives important information when solving the quadratic equation. So, as you can see, the axis of symmetry is an important concept when dealing with quadratic equations.

Finding the y-intercept: Crossing the Y-Axis

We need to determine the y-intercept of the function. The y-intercept is the point where the graph crosses the y-axis. At this point, the x-coordinate is always 0. To find the y-intercept, we substitute x = 0 into our original equation: f(x) = x² - 8x + 5. This gives us f(0) = (0)² - 8(0) + 5, which simplifies to f(0) = 5. This means the y-intercept is the point (0, 5). Since the fourth statement doesn’t provide the y-intercept, we need to verify if the y-intercept is one of the option. If the question included the point (0,5), the statement would be true. However, since the y-intercept is not an option in our case, we will have to look at the three options, which are options 1, 2, and 3. The fourth statement is not true because the value of the y-intercept is (0,5), and this value is not in any of the statements. Therefore, we should only select the first option since the vertex form is correct. Great job guys, we are almost done!

Conclusion: Wrapping it Up

So, to recap, based on our analysis, the only true statement about the graph of the function f(x) = x² - 8x + 5 is:

• The function in vertex form is f(x) = (x - 4)² - 11

We successfully transformed the equation into vertex form, identified the vertex, determined the axis of symmetry, and found the y-intercept. This exercise is a fantastic example of how understanding the different forms of a quadratic function—standard form and vertex form—can unlock a wealth of information about its graph. Keep practicing, and you'll become pros at analyzing and visualizing these functions. Keep up the awesome work, and keep exploring the amazing world of mathematics! Hope you enjoyed this mathematical journey.