Function Calculation: Solving F(x) = -2x² + X And More

Hey guys! Let's dive into some fun with functions, specifically dealing with the function f(x) = -2x² + x. We're going to calculate some values and solve an equation related to this function. Think of it as a mathematical adventure where we uncover the secrets hidden within these expressions. So, grab your thinking caps, and let’s get started!

Understanding the Function f(x) = -2x² + x

First off, it's super important to understand what this function actually means. The function f(x) = -2x² + x is a quadratic function. In the simplest terms, a quadratic function is a polynomial function of the second degree. What does that mean? Well, it means the highest power of x in the function is 2. The general form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants.

In our case, f(x) = -2x² + x, we can see that a is -2, b is 1 (since x is the same as 1x), and c is 0 (because there's no constant term added or subtracted). Quadratic functions create a parabola when graphed, which is a U-shaped curve. The coefficient a determines whether the parabola opens upwards or downwards. If a is positive, it opens upwards, and if it's negative, like in our case (-2), it opens downwards. The other coefficients, b and c, affect the position and shape of the parabola. Understanding these basics is crucial because it sets the stage for what we're about to calculate and solve.

When we talk about calculating the value of a function for a specific x, we’re essentially asking: “What is the y value when x is this number?” This gives us a point on the graph of the function. Solving f(x) for a specific value means we’re finding the x values that give us that particular y value. This is like finding the points where the function's graph intersects a horizontal line at that y value. So, let’s put this knowledge into action and start crunching some numbers!

Calculating f(0) and f(-7)

Alright, let's tackle the first part: calculating f(0) and f(-7). This is where we get to plug in some numbers and see what comes out! When we calculate f(0), we're asking, “What is the value of the function when x is 0?” It’s like checking the function’s output at the starting point.

To find f(0), we simply substitute x with 0 in the function: f(0) = -2(0)² + 0. Now, let's simplify this. First, 0 squared (0²) is 0. Then, -2 multiplied by 0 is still 0. Finally, 0 + 0 is 0. So, f(0) = 0. Easy peasy, right? This tells us that the graph of the function passes through the origin (0,0) – a fundamental point on the parabola. In practical terms, this result can signify a baseline or a starting point in scenarios where this function might model real-world phenomena.

Now, let's crank things up a notch and find f(-7). This time, we're asking, “What is the value of the function when x is -7?” This will give us another point on our parabola, potentially far from the origin. To find f(-7), we substitute x with -7 in the function: f(-7) = -2(-7)² + (-7). Remember the order of operations (PEMDAS/BODMAS): we need to deal with the exponent first. So, (-7)² is (-7) multiplied by (-7), which equals 49. Now we have f(-7) = -2(49) + (-7). Next, we multiply -2 by 49, which gives us -98. So, f(-7) = -98 + (-7). Finally, we add -98 and -7, which results in -105. Therefore, f(-7) = -105.

This result tells us that when x is -7, the function's value (or y value) is -105. This point (-7, -105) is another location on our downward-opening parabola, significantly below the x-axis. Such values can have implications in real-world contexts, such as representing a loss or a negative change if the function models profits, temperature, or other variables. Calculating these values is not just about plugging in numbers; it's about understanding how the function behaves at different points and what those points might represent in various applications.

Solving f(x) = 3x² + 2x - 5

Okay, guys, let's move on to the second part of our adventure: solving the equation f(x) = 3x² + 2x - 5. Remember, our function f(x) is -2x² + x. So, we're essentially setting up an equation where -2x² + x is equal to 3x² + 2x - 5. Think of it as a mathematical puzzle where we need to find the x values that make this equation true. These x values are the solutions to our equation, and they tell us where the two functions intersect, which is a pretty cool concept when you visualize it graphically!

The first step in solving this equation is to set everything to one side, making the equation equal to zero. Why do we do this? Well, setting an equation to zero allows us to use powerful tools like factoring or the quadratic formula to find the solutions. So, let's get to it. We start with -2x² + x = 3x² + 2x - 5. To get zero on one side, we can add 2x² and subtract x from both sides. This gives us 0 = 3x² + 2x - 5 + 2x² - x. Now, let's combine like terms. We have 3x² + 2x² which equals 5x², and 2x - x which equals x. So, our equation simplifies to 0 = 5x² + x - 5. Great! We've got a quadratic equation in the standard form: ax² + bx + c = 0, where a is 5, b is 1, and c is -5.

Now that we have our quadratic equation, we have a couple of options for solving it: factoring or using the quadratic formula. Factoring involves breaking down the quadratic expression into two binomials, which can be a quick method if the equation factors nicely. However, not all quadratic equations are easily factorable, and in this case, it doesn’t look like ours will factor neatly. So, let's bring out the big guns: the quadratic formula. The quadratic formula is a foolproof method for solving any quadratic equation, and it's given by: x = (-b ± √(b² - 4ac)) / (2a). It might look a bit intimidating, but it's really just a matter of plugging in our values for a, b, and c and simplifying.

Let's substitute our values into the formula: x = (-1 ± √(1² - 4(5)(-5))) / (2(5)). First, we simplify inside the square root. 1² is 1, and -4 times 5 times -5 is 100. So, we have x = (-1 ± √(1 + 100)) / 10. This simplifies to x = (-1 ± √101) / 10. Now, we have two possible solutions because of the ± sign. We have x = (-1 + √101) / 10 and x = (-1 - √101) / 10. To get approximate numerical values, we can calculate the square root of 101, which is roughly 10.05. So, our solutions are approximately x = (-1 + 10.05) / 10 and x = (-1 - 10.05) / 10. This gives us x ≈ 0.905 and x ≈ -1.105.

These two values are the x-coordinates where the parabola of f(x) = -2x² + x intersects the parabola of g(x) = 3x² + 2x - 5. In a real-world context, these points of intersection might represent break-even points, equilibrium states, or critical junctures in a system being modeled by these functions. So, by solving this equation, we’ve not only flexed our math muscles but also gained insights into the relationship between these two functions.

Conclusion

And there you have it! We've successfully calculated f(0) and f(-7) for the function f(x) = -2x² + x, and we've solved the equation f(x) = 3x² + 2x - 5. Awesome job, guys! We've seen how to plug values into a function to find corresponding outputs, and we've used the quadratic formula to solve a more complex equation. Remember, math isn't just about numbers and formulas; it's about understanding relationships and solving problems. These skills are incredibly valuable, whether you're dealing with abstract concepts or real-world situations. So, keep practicing, keep exploring, and most importantly, keep having fun with math!