Evaluate H(-8) For H(t) = -2(t+5)^2 + 4

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    In mathematics, a function is a fundamental concept that describes a relationship between inputs and outputs. In this article, we will delve into the evaluation of a specific function, h(t) = -2(t + 5)² + 4, at a particular input value, t = -8. Understanding function evaluation is crucial for various mathematical applications, including graphing, calculus, and modeling real-world phenomena. Before we embark on the step-by-step evaluation, let's first grasp the essence of functions and their notation.

    A function, in simple terms, is like a machine that takes an input, processes it according to a set of rules, and produces an output. The input is often represented by a variable, such as 't' in our case, and the function itself is denoted by a letter, like 'h' in 'h(t)'. The expression 'h(t)' signifies the output of the function 'h' when the input is 't'. The right-hand side of the equation, '-2(t + 5)² + 4', defines the rule or the formula that dictates how the input 't' is transformed to generate the output. Understanding this notation is paramount to successfully evaluating functions.

    Now, let's talk about evaluating a function. It means finding the output value of the function for a given input value. In our problem, we're asked to find h(-8), which means we need to determine what the function h(t) outputs when we plug in -8 for t. It's like feeding -8 into our 'h' machine and seeing what comes out. This process involves substituting the given input value into the function's formula and then performing the necessary arithmetic operations to simplify the expression. Accuracy in substitution and careful application of the order of operations are key to arriving at the correct answer. The concept of function evaluation might seem straightforward, but it is the bedrock for understanding more advanced mathematical concepts such as limits, derivatives, and integrals in calculus. It also plays a vital role in modeling real-world scenarios, where we use functions to represent relationships between different quantities.

    The task at hand is to evaluate the function h(t) = -2(t + 5)² + 4 at t = -8. This involves a systematic process of substitution and simplification, adhering to the order of operations to arrive at the correct result. Here's a breakdown of the steps involved:

    1. Substitution: The first step is to substitute the given value of 't', which is -8, into the function's formula. This means replacing every instance of 't' in the expression '-2(t + 5)² + 4' with -8. This yields the expression: h(-8) = -2((-8) + 5)² + 4. Careful substitution is crucial at this stage, as any error here will propagate through the rest of the calculation. Pay close attention to the signs and ensure that the value is placed correctly within the parentheses and exponents.

    2. Simplifying inside the parentheses: Following the order of operations (PEMDAS/BODMAS), we first simplify the expression inside the parentheses. In our case, we have (-8) + 5, which equals -3. So, the expression becomes: h(-8) = -2(-3)² + 4. This step is a straightforward arithmetic operation, but it's important to perform it accurately. A common mistake is to overlook the negative sign or to miscalculate the addition or subtraction. Remember that adding a negative number is the same as subtracting its positive counterpart.

    3. Evaluating the exponent: Next, we address the exponent. We have (-3)², which means -3 multiplied by itself. This equals 9. So, the expression now looks like: h(-8) = -2(9) + 4. It's essential to remember that squaring a negative number results in a positive number. This is because a negative times a negative equals a positive. Failing to account for this can lead to an incorrect result. Exponents indicate repeated multiplication, and understanding how they interact with negative numbers is a fundamental skill in algebra.

    4. Performing the multiplication: Now, we perform the multiplication. We have -2 multiplied by 9, which equals -18. The expression is now: h(-8) = -18 + 4. Multiplication and division take precedence over addition and subtraction in the order of operations. Therefore, we perform this step before moving on to the addition. The negative sign in front of the 2 is crucial and must be carried through the calculation correctly.

    5. Performing the addition: Finally, we perform the addition. We have -18 + 4, which equals -14. Therefore, h(-8) = -14. This is the final step in the evaluation, and it involves adding a negative number to a positive number. The result will be negative because the absolute value of the negative number is greater than the absolute value of the positive number. Ensure you understand the rules for adding numbers with different signs to arrive at the correct answer.

    To reiterate, let's go through the detailed calculation of finding h(-8) for the function h(t) = -2(t + 5)² + 4:

    1. Substitute t = -8 into the function: h(-8) = -2((-8) + 5)² + 4

    This is the foundational step where we replace the variable 't' with the specific value -8. It's a direct application of the concept of function evaluation. The substitution must be done meticulously, ensuring that the negative sign is correctly placed and that the value is enclosed in parentheses to maintain the correct order of operations.

    1. Simplify inside the parentheses: h(-8) = -2(-3)² + 4

    Here, we perform the operation within the parentheses first, according to the order of operations (PEMDAS/BODMAS). We add -8 and 5, which results in -3. This step highlights the importance of understanding integer arithmetic, particularly the addition of numbers with different signs. The negative sign is crucial and must be carried forward correctly.

    1. Evaluate the exponent: h(-8) = -2(9) + 4

    Next, we evaluate the exponent. (-3)² means -3 multiplied by itself, which equals 9. It's essential to remember that squaring a negative number results in a positive number. This step demonstrates the understanding of exponents and their interaction with negative numbers. A common mistake is to treat (-3)² as -3², which would lead to an incorrect result.

    1. Perform the multiplication: h(-8) = -18 + 4

    Now, we perform the multiplication. -2 multiplied by 9 equals -18. Multiplication takes precedence over addition in the order of operations, so we perform this step before adding 4. The negative sign is carried forward from the -2, and it's crucial to maintain this sign to ensure the correct result.

    1. Perform the addition: h(-8) = -14

    Finally, we perform the addition. -18 plus 4 equals -14. This step involves adding numbers with different signs. Since the absolute value of -18 is greater than the absolute value of 4, the result is negative. Understanding the rules for adding integers with different signs is critical for this final step. The result, -14, is the value of the function h(t) when t = -8.

    Therefore, after meticulously following each step, we arrive at the final answer: h(-8) = -14. This detailed calculation reinforces the importance of adhering to the order of operations and paying close attention to the signs of the numbers involved.

    After carefully following the steps of substitution, simplification, and arithmetic operations, we have arrived at the solution. For the function h(t) = -2(t + 5)² + 4, when t = -8, the value of the function, h(-8), is -14. This result signifies that when the input to the function 'h' is -8, the corresponding output is -14. In the context of a graph, this would represent the point (-8, -14) on the function's curve.

    In conclusion, evaluating a function at a specific input value is a fundamental skill in mathematics. It involves replacing the variable in the function's formula with the given value and then simplifying the resulting expression using the order of operations. The process requires careful attention to detail, especially when dealing with negative numbers, exponents, and parentheses. Understanding how to evaluate functions is essential for various mathematical applications, including graphing, calculus, and modeling real-world phenomena. This exercise of finding h(-8) not only provides a numerical answer but also reinforces the core principles of function evaluation, which are crucial for further mathematical studies and problem-solving.

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