Quadratic Function Equation: Vertex (-2,-4), Point (7,320)

Hey guys! Let's dive into finding the equation of a quadratic function when we're given its vertex and another point it passes through. It might sound tricky, but trust me, it's totally manageable once we break it down. We'll express our answer in the standard form, which is P(x) = ax² + bx + c. So, buckle up, and let's get started!

Understanding Quadratic Functions

Before we jump into solving the problem, let's quickly recap what quadratic functions are all about. Quadratic functions are polynomial functions of the second degree, meaning the highest power of the variable (usually 'x') is 2. The graph of a quadratic function is a parabola, a U-shaped curve that can open upwards or downwards. The standard form of a quadratic function is P(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants. The value of 'a' determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0). The vertex is a crucial point on the parabola; it's the point where the parabola changes direction – either the minimum point (if the parabola opens upwards) or the maximum point (if it opens downwards). The vertex form of a quadratic equation provides a direct way to identify the vertex, which makes it incredibly useful for this type of problem. Understanding the relationship between the vertex, the coefficients, and the overall shape of the parabola is key to mastering quadratic functions. These functions aren't just abstract math concepts; they have real-world applications in physics (projectile motion), engineering (designing arches), and even economics (modeling costs and revenue).

Using the Vertex Form

The vertex form of a quadratic function is super handy when we know the vertex. It looks like this: P(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. In our case, the vertex is given as (-2, -4), so we know that h = -2 and k = -4. Plugging these values into the vertex form, we get: P(x) = a(x - (-2))² + (-4) which simplifies to P(x) = a(x + 2)² - 4. Notice that we still have 'a' to figure out. This 'a' value is critical because it determines the parabola's stretch or compression and whether it opens upwards or downwards. The larger the absolute value of 'a', the narrower the parabola. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, it opens downwards. The vertex form not only gives us the vertex directly but also sets us up nicely to find the equation using another point on the parabola. This is because we can substitute the coordinates of the point into the equation and solve for 'a'. Understanding how the vertex form relates to the standard form can also help us convert between the two forms, giving us a more complete picture of the quadratic function.

Finding the Value of 'a'

To find the value of 'a', we'll use the other piece of information we have: the parabola passes through the point (7, 320). This means that when x = 7, P(x) = 320. Let's plug these values into our equation from the previous step: 320 = a(7 + 2)² - 4. Now we have an equation with just one unknown, 'a', which we can easily solve. First, simplify the equation: 320 = a(9)² - 4 becomes 320 = 81a - 4. Next, add 4 to both sides: 324 = 81a. Finally, divide both sides by 81 to isolate 'a': a = 324 / 81 = 4. So, we've found that a = 4. This tells us that the parabola opens upwards (since 'a' is positive) and is somewhat narrow (since 'a' is a bit larger than 1). This step is crucial because without finding 'a', we can't fully define our quadratic function. It's like having a puzzle with one piece missing – it's not complete until we find that final piece. Now that we have 'a', we're one step closer to expressing the quadratic function in the standard form.

Completing the Equation

Now that we know a = 4, we can plug it back into the vertex form: P(x) = 4(x + 2)² - 4. But remember, the question asks for the answer in the form P(x) = ax² + bx + c, so we need to expand and simplify this equation. First, let's expand the squared term: (x + 2)² = (x + 2)(x + 2) = x² + 4x + 4. Now substitute this back into our equation: P(x) = 4(x² + 4x + 4) - 4. Next, distribute the 4: P(x) = 4x² + 16x + 16 - 4. Finally, combine the constant terms: P(x) = 4x² + 16x + 12. So, we've successfully converted the equation from vertex form to standard form. This final step shows how different forms of the quadratic equation are related and how we can move between them to suit our needs. Double-checking your work at this stage is always a good idea to ensure you haven't made any arithmetic errors.

Final Answer

So, the equation of the quadratic function in the form P(x) = ax² + bx + c is: P(x) = 4x² + 16x + 12. That's it! We've found the equation that satisfies the given conditions. We started with the vertex and a point, used the vertex form to our advantage, solved for 'a', and then converted the equation to standard form. This process highlights the power of using the right form of the equation to solve problems efficiently. Remember, the key is to understand the properties of quadratic functions and how the different forms of the equation relate to these properties. With practice, you'll be able to tackle these problems with confidence.

I hope this explanation helps you guys understand how to find the equation of a quadratic function given its vertex and a point it passes through. Keep practicing, and you'll become quadratic equation pros in no time!