Find The Function: Value Table Analysis

Hey guys! Ever been faced with a table of values and asked to find the function that represents it? It might seem daunting, but don't worry, we'll break it down. In this article, we'll walk through the process step-by-step so you can tackle these problems with confidence. Let's dive in!

Understanding the Problem

So, the problem gives us a table of x and y values. Our mission, should we choose to accept it, is to figure out which of the provided functions correctly maps each x-value to its corresponding y-value. We need to test each function with the given x-values to see if the resulting y-value matches what's in the table. It's like a mathematical detective game!

Here’s the table we're working with:

And here are our function options:

a. f(x) = 5x + 19 b. f(x) = -5x - 19 c. f(x) = -5x + 19

Method 1: Testing the Functions

Option A: f(x) = 5x + 19

Let's start with the first function, f(x) = 5x + 19. We'll plug in each x-value from the table and see if we get the corresponding y-value.

• For x = 0: f(0) = 5(0) + 19 = 0 + 19 = 19. This matches the table, so far so good!
• For x = -1: f(-1) = 5(-1) + 19 = -5 + 19 = 14. Uh oh! The table says y should be 24 when x is -1. So, option A is not the correct function.

Option B: f(x) = -5x - 19

Now let's test the second function, f(x) = -5x - 19.

• For x = 0: f(0) = -5(0) - 19 = 0 - 19 = -19. This doesn't match the table (we need y = 19), so option B is out.

Option C: f(x) = -5x + 19

Finally, let's try the third function, f(x) = -5x + 19.

• For x = 0: f(0) = -5(0) + 19 = 0 + 19 = 19. Bingo! Matches the table.
• For x = -1: f(-1) = -5(-1) + 19 = 5 + 19 = 24. Perfect! Matches the table.
• For x = 2: f(2) = -5(2) + 19 = -10 + 19 = 9. Yes! Matches the table.
• For x = -2: f(-2) = -5(-2) + 19 = 10 + 19 = 29. Awesome! Matches the table.

Since option C works for all the x and y values in the table, it's the correct function.

Method 2: Using Slope and Intercept

Another way to approach this is by using the concept of slope and y-intercept. This method is particularly useful if you recognize that the function is linear (which the options suggest).

Finding the Slope

The slope (m) of a line can be found using two points (x1, y1) and (x2, y2) from the table:

m = (y2 - y1) / (x2 - x1)

Let's use the points (0, 19) and (-1, 24):

m = (24 - 19) / (-1 - 0) = 5 / -1 = -5

So, the slope is -5. This narrows down our options to functions that have -5 as the coefficient of x.

Finding the Y-Intercept

The y-intercept is the y-value when x = 0. From the table, we can see that when x = 0, y = 19. So, the y-intercept is 19.

Constructing the Function

Now we know the slope (m = -5) and the y-intercept (b = 19). The equation of a line is given by:

f(x) = mx + b

Plugging in our values:

f(x) = -5x + 19

This matches option C, confirming our earlier result.

Conclusion

So, the correct function that represents the table of values is:

c. f(x) = -5x + 19

Both methods, testing each function and using slope-intercept, lead us to the same answer. Depending on the problem and your preferences, you can choose whichever method you find easier. Keep practicing, and you'll become a pro at finding functions from tables of values! You got this!

Additional Tips for Success

• Double-Check Your Work: Always double-check your calculations, especially when dealing with negative numbers. A small mistake can lead to a wrong answer.
• Understand the Basics: Make sure you have a solid understanding of linear functions, slope, and y-intercept. This will make it easier to apply the methods discussed above.
• Practice Regularly: The more you practice, the more comfortable you'll become with these types of problems. Try to find similar problems online or in textbooks.
• Use a Calculator: Don't hesitate to use a calculator to help with calculations, especially if you're dealing with complex numbers or fractions.
• Stay Organized: Keep your work organized and clearly label each step. This will help you avoid mistakes and make it easier to review your work.

Mastering Linear Functions

Linear functions are fundamental in mathematics, and being able to identify and work with them is a crucial skill. They appear in various real-world scenarios, from calculating the cost of items to predicting trends.

Understanding the slope-intercept form (f(x) = mx + b) is key. The slope (m) tells you how much the function changes for every unit increase in x, while the y-intercept (b) tells you the value of the function when x is zero.

Being able to find the equation of a line from two points, or from a table of values, will help you solve a wide range of problems.

Common Mistakes to Avoid

• Sign Errors: Be very careful with negative signs, as they can easily lead to mistakes.
• Incorrect Slope Calculation: Make sure you subtract the y-values and x-values in the correct order when calculating the slope.
• Misidentifying the Y-Intercept: The y-intercept is the y-value when x = 0. Make sure you correctly identify it from the table or graph.
• Not Checking All Values: When testing a function, make sure you check it with all the given x-values in the table. A function might work for some values but not for others.

Real-World Applications

Understanding and working with functions, like the one we explored, has numerous real-world applications. For instance, consider a scenario where a company charges a fixed fee plus an hourly rate for their services. This can be modeled using a linear function, where the fixed fee is the y-intercept and the hourly rate is the slope.

Similarly, in physics, the relationship between distance, speed, and time can be represented using a linear function. In economics, linear functions can be used to model supply and demand curves.

By mastering the concepts and techniques discussed in this article, you'll be well-equipped to tackle a wide range of problems in mathematics and beyond. So keep practicing, stay curious, and never stop learning!