Points On Graph H(x) = \sqrt[3]{-x+2}: Find The Right Table

Have you ever been stumped by a math problem that seems to mix tables and graphs? Well, you're not alone! Today, we're going to break down a specific type of problem: identifying which table correctly represents points on the graph of the function h(x) = \sqrt[3]{-x+2}. This might sound intimidating, but don't worry – we'll take it step by step.

Understanding the Function h(x) = \sqrt[3]{-x+2}

Let's start by really understanding the function we're working with: h(x) = \sqrt[3]{-x+2}. What does this mean? This is a cube root function. The \sqrt[3]{} symbol means we're looking for a number that, when multiplied by itself three times, equals the value inside the root. For instance, the cube root of 8 is 2 because 2 * 2 * 2 = 8.

Now, let's look at the expression inside the cube root: -x + 2. This means we're taking the negative of x, adding 2, and then finding the cube root of the result. This function takes an input 'x', performs some math on it, and gives us an output, which we call h(x) or 'y'. The key here is to understand that each 'x' value will correspond to a specific 'y' value based on this function. Think of it like a machine: you put in 'x', and the machine spits out 'y'. Our goal is to find a table where the 'x' and 'y' values match the results we'd get from this function.

When we're trying to find points on the graph of this function, we're essentially looking for pairs of (x, y) values that satisfy the equation. Each of these pairs represents a point on the graph. A table that correctly represents the function will list several of these points. To determine if a table is correct, we need to test the x-values in the table by plugging them into the function h(x) = \sqrt[3]{-x+2} and checking if the resulting y-values match the values listed in the table. This process involves careful calculation and a solid understanding of cube roots and algebraic manipulation.

So, to reiterate, the core of this task lies in plugging in x-values from the table into our function and verifying that we obtain the corresponding y-values. This is a fundamental concept in algebra and is crucial for understanding how functions behave and how they are represented graphically. The cube root adds an interesting twist, as it's neither a square root nor a simple linear function, but it still adheres to the same basic principles of function evaluation.

How to Check if a Table Represents the Function

So, how do we actually check if a table represents the function h(x) = \sqrt[3]{-x+2}? It's simpler than it looks. We need to take each 'x' value from the table, plug it into the function, and see if we get the corresponding 'y' value. Let's break this down into a step-by-step process:

1. Choose an x-value from the table: Start with the first x-value in the table. For example, if the table includes x = -8, that's where we begin.
2. Substitute the x-value into the function: Replace 'x' in the equation h(x) = \sqrt[3]{-x+2} with the chosen x-value. So, if we're using x = -8, the equation becomes h(-8) = \sqrt[3]{-(-8)+2}.
3. Simplify the expression: Now, we need to simplify the expression inside the cube root. Remember the order of operations (PEMDAS/BODMAS). In our example, -(-8) becomes +8, so we have h(-8) = \sqrt[3]{8+2}, which simplifies to h(-8) = \sqrt[3]{10}.
4. Calculate the cube root: Find the cube root of the simplified value. The cube root of 10 is approximately 2.15. However, we need to be precise. If the table has a y-value that doesn't match this (or the precise calculation), then this point does not lie on the graph of the function.
5. Compare the calculated y-value with the table's y-value: Look at the table and see what y-value corresponds to the x-value we used. If the calculated h(x) value (our 'y' value) matches the y-value in the table, then this point is likely on the graph. If they don't match, the table is incorrect.
6. Repeat for all x-values in the table: We need to repeat steps 1-5 for every x-value in the table. If even one x-value doesn't produce the correct y-value, the entire table is incorrect for this function.

This methodical approach ensures that we're accurately verifying each point. It's crucial to remember that even if most of the points in the table seem to fit, a single incorrect point invalidates the entire table as a representation of the function. Accuracy and attention to detail are key in these calculations. Each calculation is a mini-investigation, making sure that the values align with the function's rule.

Example Walkthrough

Let's take a look at a practical example using the function h(x) = \sqrt[3]{-x+2} and one of the tables provided. This will make the process even clearer. Suppose we have the following table:

We'll go through each x-value one by one:

• x = -8:
  • h(-8) = \sqrt[3]{-(-8)+2} = \sqrt[3]{8+2} = \sqrt[3]{10} ≈ 2.15
  • The table shows y = 4. This does NOT match our calculated value, so the point (-8, 4) is not on the graph.
• x = -1:
  • h(-1) = \sqrt[3]{-(-1)+2} = \sqrt[3]{1+2} = \sqrt[3]{3} ≈ 1.44
  • The table shows y = 3. Again, this does NOT match, so the point (-1, 3) is not on the graph.

Since we've already found two points that don't match, we can confidently say that this table does not correctly represent the function h(x) = \sqrt[3]{-x+2}. We wouldn't even need to check the remaining x-values in this case. This illustrates the efficiency of the process - as soon as we find a mismatch, we can disqualify the table.

Let's imagine we were checking another table, and the first few points matched. We would continue meticulously, point by point, until we either found a mismatch or confirmed that all the points on the table accurately represent the function. This example underscores the importance of careful computation and the systematic approach we discussed earlier.

Common Mistakes to Avoid

When working with functions and tables, it's easy to make mistakes if you're not careful. Let's go over some common pitfalls to avoid when determining if a table represents the function h(x) = \sqrt[3]{-x+2}:

1. Incorrect Order of Operations: This is a big one! Always remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Make sure you perform operations in the correct order. For example, in h(x) = \sqrt[3]{-x+2}, you need to deal with the negative sign before adding 2.
2. Sign Errors: Pay very close attention to signs, especially when dealing with negative numbers. A simple sign error can throw off the entire calculation. In our function, the '-x' term is a prime area for mistakes. For instance, -(-3) is +3, not -3.
3. Miscalculating Cube Roots: Cube roots can be tricky if you're trying to calculate them by hand. If you're allowed to use a calculator, double-check your inputs. If not, try to simplify the expression inside the cube root as much as possible before attempting to find the root. It helps to know some common cube roots like \sqrt[3]{8} = 2 and \sqrt[3]{27} = 3.
4. Stopping Too Early: As we mentioned earlier, you need to check every x-value in the table. Don't assume the table is correct just because the first few points match. A single incorrect point invalidates the whole table.
5. Approximation Errors: When dealing with cube roots that are not whole numbers, you'll often need to approximate. Be careful with rounding. It's best to keep as much precision as possible until the very end of the calculation. Slight rounding errors can lead to a mismatch between the calculated y-value and the table's y-value.
6. Misinterpreting the Function: Make sure you fully understand what the function is asking you to do. With h(x) = \sqrt[3]{-x+2}, it's essential to recognize that you're finding the cube root of the entire expression -x + 2, not just the cube root of -x, and then adding 2. Understanding the scope of operations is critical.

By being aware of these common mistakes and taking steps to avoid them, you'll significantly increase your accuracy and confidence when working with functions and tables.

Conclusion

Determining if a table represents a function like h(x) = \sqrt[3]{-x+2} might seem daunting at first, but by breaking it down into a series of clear, manageable steps, it becomes a much more approachable task. Remember to carefully substitute x-values, simplify expressions, calculate cube roots, and compare your results with the table. Most importantly, be meticulous and avoid common mistakes. With practice, you'll become a pro at identifying points on graphs and understanding how functions work.

For further exploration and practice with functions and their graphs, consider visiting Khan Academy's Algebra I section. You'll find tons of resources and exercises to help you master these concepts!