Analyzing F(x) = 4x² - X - 3: Point And Value Evaluation

In this article, we'll dive into the function f(x) = 4x² - x - 3. We'll tackle two key questions: first, whether a specific point lies on the graph of this function, and second, how to evaluate the function at a given value of x. Understanding these concepts is crucial for mastering functions in mathematics. Let's break it down step by step.

(a) Is the Point (-1, 2) on the Graph of f?

To determine if the point (-1, 2) lies on the graph of the function f(x) = 4x² - x - 3, we need to check if the function's output matches the y-coordinate of the point when we plug in the x-coordinate. In other words, we substitute x = -1 into the function and see if we get f(-1) = 2.

Let's calculate f(-1):

f(-1) = 4(-1)² - (-1) - 3

First, we square -1:

(-1)² = 1

Next, we multiply by 4:

4 * 1 = 4

Then, we handle the subtraction of -1, which becomes addition:

4 - (-1) = 4 + 1 = 5

Finally, we subtract 3:

5 - 3 = 2

So, f(-1) = 2. This means that when x = -1, the function's output is indeed 2. Therefore, the point (-1, 2) does lie on the graph of the function f(x) = 4x² - x - 3. When dealing with functions and graphs, always remember that a point lies on the graph if and only if substituting the x-coordinate into the function yields the y-coordinate. This is a fundamental concept in coordinate geometry and function analysis. Understanding this relationship allows us to visualize and analyze functions effectively, making it easier to solve problems related to function behavior and graphical representation. So, in summary, plugging in x = -1 into the function gives us y = 2, confirming that the point (-1, 2) is indeed on the graph of f(x).

(b) If x = 2, What is f(x)?

Now, let's evaluate the function f(x) = 4x² - x - 3 when x = 2. This means we need to substitute x = 2 into the function and calculate the resulting value. This process will give us the function's output at that specific x-value.

So, we want to find f(2):

f(2) = 4(2)² - (2) - 3

First, we square 2:

(2)² = 4

Next, we multiply by 4:

4 * 4 = 16

Then, we subtract 2:

16 - 2 = 14

Finally, we subtract 3:

14 - 3 = 11

Thus, f(2) = 11. This tells us that when x = 2, the function's value is 11. In graphical terms, this means the point (2, 11) lies on the graph of the function f(x) = 4x² - x - 3. Evaluating functions at specific points is a common task in mathematics and has many applications. For example, it can help us find the height of a curve at a certain point, determine the output of a model given a specific input, or analyze the behavior of a function in different intervals. When evaluating functions, always follow the order of operations (PEMDAS/BODMAS) to ensure accurate results. This involves dealing with parentheses, exponents, multiplication and division, and addition and subtraction in the correct sequence. Practice evaluating functions with different types of expressions to build your skills and confidence in this area. Remember, the key is to substitute the given value for the variable and simplify the expression carefully, paying attention to signs and operations. In this case, we found that f(2) = 11, which gives us a specific point on the function's graph and provides valuable information about the function's behavior.

In summary, by substituting x = 2 into the function, we find that the corresponding y-value is 11. This is a straightforward application of function evaluation, which is a fundamental skill in mathematics.

Conclusion

Alright, guys, we've successfully analyzed the function f(x) = 4x² - x - 3. We determined that the point (-1, 2) does indeed lie on the graph of the function, and we found that f(2) = 11. These exercises highlight the importance of understanding how to evaluate functions and interpret their graphical representation. Remember, functions are a fundamental concept in mathematics, and mastering them will open doors to more advanced topics and applications. Keep practicing, and you'll become a pro in no time! Understanding function evaluation and graph interpretation is key to success in many areas of math. Whether it's determining if a point lies on a curve or calculating the output for a given input, these skills are invaluable. By breaking down the problem into smaller, manageable steps, you can tackle even the most complex functions with confidence. So, don't be afraid to dive in and explore the fascinating world of functions!

In conclusion, understanding how to work with functions like f(x) = 4x² - x - 3 is super useful. Knowing that (-1, 2) is on the graph and that f(2) = 11 gives us a solid grasp of how the function behaves. Keep practicing with different functions, and you'll become more confident and skilled in your math journey. Always remember to substitute carefully and follow the order of operations, and you'll be able to solve a wide range of problems related to functions and their graphs. With consistent effort, you'll unlock the full potential of functions and their applications in various fields. Keep going, and you'll become a math whiz!