Hey everyone! Today, we're diving deep into the world of polynomial functions and their graphs. Specifically, we're going to dissect the function f(x) = 0.03x²(x² - 25) and figure out what its graph looks like. This isn't just about blindly memorizing rules; it's about understanding the relationship between a function's equation and its visual representation. So, buckle up, grab your thinking caps, and let's get started!
Understanding the Function: f(x) = 0.03x²(x² - 25)
Before we even think about the graph, let's break down the function itself. The function f(x) = 0.03x²(x² - 25) is a polynomial function. Polynomial functions are those that involve only non-negative integer powers of x. They're smooth, continuous curves, which makes them relatively predictable – a huge advantage when we're trying to sketch their graphs. The key characteristic to note here is that this is a quartic function (degree 4) because the highest power of x when you expand the function will be x⁴. The coefficient of this term will be 0.03, which is positive. This tells us that as x approaches positive or negative infinity, the function will also approach positive infinity. In simpler terms, the graph will rise on both the left and right ends. Factoring the function is also a crucial step. We can see the function is already partially factored as f(x) = 0.03x²(x² - 25). We can further factor the expression (x² - 25) using the difference of squares formula: a² - b² = (a - b)(a + b). Applying this, we get x² - 25 = (x - 5)(x + 5). Therefore, the fully factored form of our function is f(x) = 0.03x²(x - 5)(x + 5). This factored form is incredibly useful because it directly reveals the roots (or zeros) of the function, which are the points where the graph intersects the x-axis. The zeros are the values of x that make f(x) = 0. From our factored form, we can see that the zeros are x = 0, x = 5, and x = -5. But here’s a crucial detail: the factor x² means that the root x = 0 has a multiplicity of 2. This multiplicity affects how the graph behaves at that x-intercept. When a root has an even multiplicity, the graph touches the x-axis at that point but doesn't cross it. When a root has an odd multiplicity, the graph crosses the x-axis. So, at x = 0, the graph will touch the x-axis and turn around, while at x = 5 and x = -5, the graph will cross the x-axis. The coefficient 0.03 is a vertical stretch factor. Since it's a positive number less than 1, it means the graph will be stretched vertically but also compressed towards the x-axis compared to the graph of x⁴. This will make the curve less steep. By carefully analyzing the function's equation, zeros, multiplicities, and leading coefficient, we've gathered essential information that will guide us in sketching or identifying its graph.
Key Features of the Graph
Alright, let's nail down the key features that will help us spot the correct graph. Understanding these elements is like having a roadmap for our graphical journey. First up are the x-intercepts, also known as the roots or zeros of the function. As we figured out earlier, our function f(x) = 0.03x²(x² - 25) has x-intercepts at x = -5, x = 0, and x = 5. These are the points where the graph crosses or touches the x-axis. Remember, the x-intercepts are the solutions to the equation f(x) = 0. They are the values of x that make the function equal to zero. The x-intercepts are crucial landmarks on the graph, and we can immediately eliminate any graph that doesn't have these intercepts. Next, we have the behavior at the x-intercepts. This is where the concept of multiplicity comes into play. We determined that x = 0 is a root with multiplicity 2, while x = -5 and x = 5 are roots with multiplicity 1. At x = 0 (multiplicity 2), the graph will touch the x-axis and bounce back, creating a turning point. At x = -5 and x = 5 (multiplicity 1), the graph will pass straight through the x-axis. This touching and crossing behavior is a powerful visual cue for identifying the correct graph. Another important feature is the end behavior of the graph. This refers to what happens to the function's values (f(x)) as x approaches positive infinity (x → ∞) and negative infinity (x → -∞). Since our function is a quartic (degree 4) polynomial with a positive leading coefficient (0.03), the end behavior is as follows: as x → ∞, f(x) → ∞, and as x → -∞, f(x) → ∞. In simple terms, the graph rises on both the left and right ends. This means that as we move further and further away from the origin in either direction along the x-axis, the y-values of the function become larger and larger. The y-intercept is the point where the graph intersects the y-axis. This occurs when x = 0. Plugging x = 0 into our function, we get f(0) = 0.03(0)²((0)² - 25) = 0. So, the y-intercept is at (0, 0), which is the same as one of our x-intercepts. The symmetry of the graph is another helpful feature. Polynomial functions can exhibit symmetry about the y-axis (even functions) or symmetry about the origin (odd functions). Our function f(x) = 0.03x²(x² - 25) is an even function because all the powers of x are even (x² and x⁴ when expanded). Even functions are symmetric about the y-axis, meaning if you fold the graph along the y-axis, the two halves will match perfectly. The number of turning points (local maxima and minima) can also provide clues. A polynomial of degree n can have at most n - 1 turning points. Our function is of degree 4, so it can have at most 3 turning points. By piecing together these key features – x-intercepts, behavior at x-intercepts, end behavior, y-intercept, symmetry, and the potential number of turning points – we create a comprehensive profile of the graph we're looking for.
Process of Elimination: Finding the Right Graph
Now that we're armed with a solid understanding of the function's properties and the key features of its graph, let's talk strategy. When faced with multiple graph options, the process of elimination is your best friend. It's like detective work – we use clues to rule out suspects until we're left with the culprit, or in this case, the correct graph. The first step is to check the x-intercepts. Look for the points where the graph crosses or touches the x-axis. Remember, our function f(x) = 0.03x²(x² - 25) has x-intercepts at x = -5, x = 0, and x = 5. Immediately eliminate any graphs that don't have these x-intercepts. This simple check can often narrow down the choices significantly. Next, scrutinize the behavior at the x-intercepts. Does the graph cross the x-axis at x = -5 and x = 5? Does it touch the x-axis and turn around at x = 0? If a graph doesn't exhibit this specific behavior, it's not the correct graph. This is where understanding the concept of multiplicity becomes crucial. If you see a graph crossing at an intercept where it should be touching, or vice versa, you can eliminate it. Pay close attention to the end behavior. Does the graph rise on both the left and right ends? If a graph falls on either end, it's not a match for our function, which has a positive leading coefficient and even degree. End behavior is a quick and easy way to eliminate graphs that have the wrong overall shape. Check the y-intercept. While this might seem redundant since we know the y-intercept is (0, 0), it's another point of confirmation. If a graph has a different y-intercept, it's not the right one. Assess the symmetry of the graph. Is the graph symmetric about the y-axis? If a graph is not symmetric about the y-axis, it can be eliminated because our function is even. Symmetry is a powerful visual indicator that can help you quickly identify the correct graph. Count the turning points. Our quartic function can have at most 3 turning points. If a graph has more than 3 turning points, it's not the graph of our function. The number of turning points provides a constraint that can help you rule out incorrect options. By systematically applying these elimination criteria, you can work your way through the graph options and confidently identify the one that matches all the key features of the function f(x) = 0.03x²(x² - 25).
Putting It All Together: Visualizing the Graph
Let's bring everything we've discussed together and paint a mental picture of what the graph of f(x) = 0.03x²(x² - 25) looks like. This visualization will be the ultimate test of our understanding. We know the graph has x-intercepts at x = -5, x = 0, and x = 5. At x = -5 and x = 5, the graph crosses the x-axis, indicating single roots. At x = 0, the graph touches the x-axis and turns around, indicating a root with multiplicity 2. This touching behavior is a key visual cue. Since the function is a quartic polynomial with a positive leading coefficient, the end behavior is that the graph rises on both the left and right ends. This means the graph extends upwards as we move away from the origin in either direction along the x-axis. The graph is symmetric about the y-axis because the function is even. This symmetry means that the left and right halves of the graph are mirror images of each other. We know the y-intercept is at (0, 0), which coincides with one of the x-intercepts. This point is where the graph touches the x-axis and changes direction. The graph can have at most 3 turning points. Considering the x-intercepts and the end behavior, we can expect to see a local maximum between x = -5 and x = 0, a local minimum at x = 0, and another local maximum between x = 0 and x = 5. Visualizing these turning points helps us understand the overall shape of the graph. Putting all these features together, we can imagine a graph that starts high on the left, comes down and crosses the x-axis at x = -5, rises to a local maximum, comes down and touches the x-axis at x = 0, turns around and rises to another local maximum, comes down and crosses the x-axis at x = 5, and then continues to rise on the right. This mental image is a powerful tool for recognizing the correct graph when presented with options. It's not just about memorizing features; it's about connecting those features to form a cohesive visual representation of the function. By practicing this visualization process, you'll become more adept at quickly and accurately identifying graphs of polynomial functions.
Conclusion
So there you have it! We've thoroughly explored the function f(x) = 0.03x²(x² - 25) and uncovered the secrets to identifying its graph. From understanding the significance of zeros and multiplicities to recognizing end behavior and symmetry, we've equipped ourselves with a comprehensive toolkit for tackling polynomial graphs. Remember, graphing functions isn't just about plotting points; it's about understanding the interplay between the equation and its visual representation. By breaking down the function, identifying key features, and using the process of elimination, you can confidently decipher the graphs of even the most complex polynomial functions. Keep practicing, and you'll become a graph-decoding master in no time! Keep exploring and happy graphing!