Max Value Of Quadratic Function: F(x)=-x^2-2x+3

Hey guys! Today, we're diving deep into the fascinating world of quadratic functions, specifically how to pinpoint their maximum value. We'll be tackling the equation $f(x) = -x^2 - 2x + 3$, breaking it down step by step so you can confidently conquer any similar problem. So, buckle up, and let's get started!

Understanding Quadratic Functions

Before we jump into the specifics, let's establish a solid foundation. Quadratic functions are polynomial functions of the second degree, meaning the highest power of the variable (usually 'x') is 2. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where 'a', 'b', and 'c' are constants. These constants play crucial roles in determining the shape and position of the parabola, which is the graphical representation of a quadratic function.

The coefficient 'a' is particularly important. It dictates whether the parabola opens upwards or downwards. If 'a' is positive, the parabola opens upwards, resembling a 'U' shape. This means the function has a minimum value at its vertex. Conversely, if 'a' is negative, the parabola opens downwards, resembling an upside-down 'U' shape. In this case, the function has a maximum value at its vertex. Since our equation, $f(x) = -x^2 - 2x + 3$, has a negative 'a' value (-1), we know we're dealing with a parabola that opens downwards and thus has a maximum value.

The vertex, my friends, is the turning point of the parabola. It's the point where the function transitions from decreasing to increasing (for upward-opening parabolas) or from increasing to decreasing (for downward-opening parabolas). For a downward-opening parabola, like the one we're working with, the vertex represents the highest point on the graph, hence the maximum value of the function. To find this maximum value, we need to determine the coordinates of the vertex.

Now, there are a couple of ways to find the vertex. We can use the vertex formula or complete the square. Let's explore both methods to give you a comprehensive understanding.

Method 1: The Vertex Formula

The vertex formula is a handy tool that directly provides the coordinates of the vertex. For a quadratic function in the form $f(x) = ax^2 + bx + c$, the x-coordinate of the vertex (often denoted as 'h') is given by:

$$h = -b / 2a$$

Once we find 'h', we can substitute it back into the original equation to find the y-coordinate of the vertex (often denoted as 'k'), which represents the maximum or minimum value of the function. So, $k = f(h)$.

Let's apply this to our equation, $f(x) = -x^2 - 2x + 3$. Here, 'a' is -1, and 'b' is -2. Plugging these values into the vertex formula, we get:

$$h = -(-2) / (2 * -1) = 2 / -2 = -1$$

So, the x-coordinate of the vertex is -1. Now, let's find the y-coordinate by substituting h = -1 back into the function:

$$k = f(-1) = -(-1)^2 - 2(-1) + 3 = -1 + 2 + 3 = 4$$

Therefore, the vertex of the parabola is (-1, 4). Since the parabola opens downwards, the y-coordinate of the vertex, which is 4, represents the maximum value of the function.

Method 2: Completing the Square

Another powerful technique for finding the vertex is completing the square. This method involves rewriting the quadratic function in vertex form, which directly reveals the vertex coordinates. The vertex form of a quadratic function is:

$$f(x) = a(x - h)^2 + k$$

Where (h, k) represents the vertex of the parabola.

To complete the square, we need to manipulate our original equation, $f(x) = -x^2 - 2x + 3$, to match this form. Here's how it works:

1. Factor out the coefficient of the x² term (which is -1 in our case) from the first two terms:

$$f(x) = -(x^2 + 2x) + 3$$

2. Complete the square inside the parentheses. To do this, take half of the coefficient of the x term (which is 2), square it (1), and add and subtract it inside the parentheses. This maintains the equality of the equation:

$$f(x) = -(x^2 + 2x + 1 - 1) + 3$$

3. Rewrite the expression inside the parentheses as a squared term:

$$f(x) = -((x + 1)^2 - 1) + 3$$

4. Distribute the negative sign and simplify:

$$f(x) = -(x + 1)^2 + 1 + 3$$

$$f(x) = -(x + 1)^2 + 4$$

Now, our equation is in vertex form! Comparing it to $f(x) = a(x - h)^2 + k$, we can see that h = -1 and k = 4. So, the vertex is (-1, 4), and the maximum value of the function is 4, just like we found using the vertex formula.

Visualizing the Parabola

To solidify our understanding, let's visualize the parabola. We know it opens downwards because 'a' is negative. We've also found the vertex to be (-1, 4). This means the highest point on the graph is at the coordinates (-1, 4). We can also find the x-intercepts (where the parabola crosses the x-axis) by setting f(x) = 0 and solving for x:

$$0 = -x^2 - 2x + 3$$

We can factor this quadratic equation:

$$0 = -(x^2 + 2x - 3)$$

$$0 = -(x + 3)(x - 1)$$

This gives us x-intercepts at x = -3 and x = 1. Knowing the vertex and the x-intercepts allows us to sketch a pretty accurate graph of the parabola. You'll see that it's a symmetrical curve with its peak at the vertex (-1, 4), confirming that 4 is indeed the maximum value.

Key Takeaways and Practice Problems

So, guys, we've covered a lot! Let's recap the key takeaways:

• Quadratic functions have the form $f(x) = ax^2 + bx + c$ and their graphs are parabolas.
• The sign of 'a' determines whether the parabola opens upwards (a > 0) or downwards (a < 0).
• The vertex is the turning point of the parabola and represents the maximum or minimum value of the function.
• We can find the vertex using the vertex formula ($h = -b / 2a$ and $k = f(h)$) or by completing the square.

To truly master this, practice is key! Here are a few practice problems for you to try:

1. Find the maximum value of the function $f(x) = -2x^2 + 8x - 5$.
2. Determine the minimum value of the function $f(x) = x^2 - 4x + 7$.
3. A ball is thrown upwards, and its height (in meters) after 't' seconds is given by the function $h(t) = -5t^2 + 20t$. What is the maximum height the ball reaches?

Work through these problems using the methods we discussed. Don't be afraid to make mistakes – that's how we learn! And remember, understanding quadratic functions is a fundamental concept in mathematics, with applications in various fields like physics, engineering, and economics.

Conclusion

Finding the maximum value of a quadratic function is a valuable skill. Whether you prefer the vertex formula or completing the square, the ability to identify the vertex is crucial. By understanding the properties of parabolas and mastering these techniques, you'll be well-equipped to tackle any quadratic function problem that comes your way. Keep practicing, keep exploring, and keep learning! You guys got this!