Vertex Form: Find The Missing Value In Function F(x)

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Hey guys! Let's dive into the world of quadratic functions and vertex form. Today, we're tackling a problem where we need to find a missing value in the process of converting a quadratic function into vertex form. This is a super important skill for understanding the behavior of parabolas and solving various math problems. So, let's get started!

Understanding Vertex Form

Before we jump into the problem, let's quickly recap what vertex form is all about. The vertex form of a quadratic function is expressed as:

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

Where:

  • a determines the direction and stretch of the parabola.
  • (h, k) represents the vertex of the parabola, which is the point where the parabola changes direction (either the minimum or maximum point).

The vertex form is incredibly useful because it directly reveals the vertex of the parabola, making it easier to graph the function and analyze its properties. Converting a quadratic function from standard form to vertex form often involves a process called completing the square, which we'll see in action shortly.

The Problem: Finding the Missing Value

Our mission today is to find a missing value in the steps of writing the function f(x) = 18x + 3x^2 in vertex form. The steps provided show the initial transformations, but there's a gap we need to fill. Let's take a look at the given steps:

  1. Write the function in standard form: f(x) = 3x^2 + 18x
  2. Factor out a from the first two terms: f(x) = 3(x^2 + 6x)
  3. Form a perfect square

Notice that the last step is incomplete. We need to figure out how to "form a perfect square" and what value is missing in the subsequent transformation. This is where the technique of completing the square comes into play.

Step-by-Step Solution: Completing the Square

Let's break down the process of completing the square to find the missing value and express the function in vertex form.

Step 1: Write the function in standard form.

The first step is already done for us. We have:

f(x) = 3x^2 + 18x

This is the standard form of a quadratic function, where the terms are arranged in descending order of their exponents.

Step 2: Factor out a from the first two terms.

The second step is also provided:

f(x) = 3(x^2 + 6x)

Here, we factored out the coefficient of the x^2 term, which is 3. This is a crucial step in completing the square because it isolates the quadratic and linear terms inside the parentheses.

Step 3: Complete the square.

This is where we need to do some work. To complete the square, we need to add and subtract a value inside the parentheses that will create a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form (x + b)^2 or (x - b)^2.

To find the value we need to add and subtract, we take half of the coefficient of the x term (which is 6 in this case), square it, and then add and subtract it inside the parentheses.

  • Half of 6 is 3.
  • 3 squared is 9.

So, we add and subtract 9 inside the parentheses:

f(x) = 3(x^2 + 6x + 9 - 9)

Step 4: Rewrite as a perfect square trinomial.

Now, we can rewrite the first three terms inside the parentheses as a perfect square trinomial:

f(x) = 3((x + 3)^2 - 9)

The expression x^2 + 6x + 9 is a perfect square trinomial because it can be factored as (x + 3)^2. This is the key to transforming the function into vertex form.

Step 5: Distribute and simplify.

Next, we distribute the 3 back into the parentheses:

f(x) = 3(x + 3)^2 - 3(9)

f(x) = 3(x + 3)^2 - 27

Step 6: Identify the vertex form.

Now, our function is in vertex form:

f(x) = 3(x + 3)^2 - 27

Comparing this to the general vertex form f(x) = a(x - h)^2 + k, we can identify the values:

  • a = 3
  • h = -3 (Note the sign change because of the (x - h) form)
  • k = -27

Therefore, the vertex of the parabola is (-3, -27).

The Missing Value

Looking back at the original steps, the missing value was the constant term that completed the square. In our case, that value was 9, which we added and subtracted inside the parentheses.

So, the complete steps are:

  1. Write the function in standard form: f(x) = 3x^2 + 18x
  2. Factor out a from the first two terms: f(x) = 3(x^2 + 6x)
  3. Form a perfect square: f(x) = 3(x^2 + 6x + 9 - 9)

Key Takeaways

  • Completing the square is a powerful technique for converting quadratic functions into vertex form.
  • The vertex form f(x) = a(x - h)^2 + k directly reveals the vertex of the parabola at the point (h, k).
  • To complete the square, take half of the coefficient of the x term, square it, and add and subtract it inside the parentheses.

Why is Vertex Form Important?

Understanding vertex form is crucial for several reasons:

  1. Graphing Parabolas: Vertex form makes it incredibly easy to graph a parabola. You can immediately identify the vertex, which is the turning point of the parabola. The a value tells you whether the parabola opens upwards (if a > 0) or downwards (if a < 0) and how stretched or compressed it is.
  2. Finding Maximum or Minimum Values: The vertex represents the maximum or minimum value of the quadratic function. If the parabola opens downwards (a < 0), the vertex is the maximum point. If it opens upwards (a > 0), the vertex is the minimum point. This is super useful in optimization problems where you need to find the highest or lowest value of a function.
  3. Solving Quadratic Equations: Completing the square, which is the technique used to get to vertex form, is also a fundamental method for solving quadratic equations. It allows you to rewrite the equation in a form where you can easily isolate the variable.
  4. Real-World Applications: Quadratic functions and parabolas pop up in various real-world scenarios, such as projectile motion (the path of a ball thrown in the air), the shape of suspension bridges, and the design of satellite dishes. Understanding vertex form helps you analyze and model these situations.

Practice Makes Perfect

Now that we've walked through this problem, the best way to solidify your understanding is to practice! Try converting different quadratic functions into vertex form and identifying their vertices. You can also try solving problems where you're given the vertex and a point on the parabola and asked to find the equation of the function.

Conclusion

So, there you have it! We successfully found the missing value in the steps of writing the function f(x) = 18x + 3x^2 in vertex form. By understanding the process of completing the square and the significance of vertex form, you're now better equipped to tackle quadratic function problems. Keep practicing, and you'll become a vertex form pro in no time! Remember, math can be fun, especially when you break it down step by step. Keep exploring, keep learning, and keep those math muscles flexing!