Find Missing Values In Function Table: F(x) = 2x^2 - 8x + 6

Let's dive into the world of functions and tables! In this article, we'll tackle the problem of finding missing values in a function table. Specifically, we'll focus on the function f(x) = 2x² - 8x + 6. You might encounter this type of problem in math class, on a test, or even in real-world applications where you need to understand how inputs and outputs relate. We'll break down the process step-by-step so you can confidently solve similar problems.

Understanding the Problem

First, let's look at the table we need to complete:

Our mission is to figure out the values of x, y, and z. Remember, in a function, we plug in a value for x and get a corresponding value for f(x). The table shows us some pairs of x and f(x) values, and we'll use the function f(x) = 2x² - 8x + 6 to find the missing ones.

Calculating the Missing Values

Finding 'x'

Oops! It seems there's a slight mix-up in the table. The x in the f(x) column when x = 0 should actually be the result of plugging 0 into the function. Let's fix that. To find the correct value, we'll substitute x = 0 into our function:

f(0) = 2(0)² - 8(0) + 6

Let's simplify this:

f(0) = 2(0) - 0 + 6 f(0) = 0 - 0 + 6 f(0) = 6

So, the first missing value, which was incorrectly labeled as x, is actually 6. Now our table looks like this:

Finding 'y'

Next, we need to find the value of y. y is the value of the function when x = 2. So, we'll substitute x = 2 into our function:

f(2) = 2(2)² - 8(2) + 6

Let's break it down:

f(2) = 2(4) - 16 + 6 f(2) = 8 - 16 + 6 f(2) = -8 + 6 f(2) = -2

Therefore, y = -2. We're making progress!

Finding 'z'

Now, let's find z. z represents the value of the function when x = 3. We'll do the same thing – substitute x = 3 into our function:

f(3) = 2(3)² - 8(3) + 6

Let's simplify:

f(3) = 2(9) - 24 + 6 f(3) = 18 - 24 + 6 f(3) = -6 + 6 f(3) = 0

So, z = 0. We've found all the missing values!

The Completed Table

Here's our completed table:

We've successfully calculated all the missing values: the value corresponding to x = 0 is 6, y = -2, and z = 0. Great job!

Key Takeaways

• Function Notation: Remember that f(x) means "the value of the function f at x". We substitute the value of x into the function's equation to find the corresponding f(x) value.
• Order of Operations: Always follow the order of operations (PEMDAS/BODMAS) when simplifying expressions. This ensures you get the correct answer.
• Careful Calculation: Double-check your calculations to avoid mistakes. A small error can throw off the entire result.
• Practice Makes Perfect: The more you practice these types of problems, the easier they become. Try working through similar examples to build your confidence.

Expanding Your Understanding

Graphing the Function

It can be helpful to visualize the function f(x) = 2x² - 8x + 6. This function is a quadratic, which means its graph is a parabola (a U-shaped curve). The table we completed gives us several points on this parabola: (0, 6), (1, 0), (2, -2), (3, 0), and (4, 6). You can plot these points on a graph and connect them to see the shape of the parabola. The graph can provide a visual confirmation of our calculations and give you a better understanding of the function's behavior.

Finding the Vertex

The vertex of a parabola is the point where it changes direction (either the lowest point or the highest point). For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by the formula x = -b / 2a. In our case, a = 2 and b = -8, so the x-coordinate of the vertex is:

x = -(-8) / (2 * 2) = 8 / 4 = 2

Notice that this is the same x-value where we found y = -2. This makes sense because the vertex is the minimum point of this parabola (since a is positive), and our table shows that f(2) = -2 is the lowest value in the table.

Applications of Quadratic Functions

Quadratic functions have many real-world applications. They can be used to model the trajectory of a projectile (like a ball thrown in the air), the shape of a suspension bridge cable, and even the profit of a business. Understanding how to work with quadratic functions, including finding values in a table, is a valuable skill in many fields.

Practice Problems

To solidify your understanding, try these practice problems:

1. Complete the table for the function g(x) = x² - 4x + 3:

   x	g(x)
   0
   1
   2
   3
   4

2. Find the missing values in the table for the function h(x) = -x² + 6x - 5:

   x	h(x)
   0
   1
   	3
   3
   4

Conclusion

We've successfully navigated the process of finding missing values in a function table! By substituting the given x values into the function and simplifying, we were able to determine the corresponding f(x) values. Remember to always pay attention to the order of operations and double-check your calculations. With practice, you'll become a pro at working with functions and tables. For further learning, you can explore more about functions and their graphs on resources like Khan Academy's Algebra I course. Keep practicing, and you'll continue to build your math skills!