Finding The Range Of Y = -2x^4 + 7 A Comprehensive Guide
Introduction
In the realm of mathematics, understanding the range of a function is crucial for grasping its behavior and limitations. This article delves into determining the range of the quartic function y = -2x⁴ + 7. We'll explore the characteristics of quartic functions, focusing on how the leading coefficient and constant term influence the range. By the end of this discussion, you'll not only understand the range of this specific function but also gain insights into analyzing the range of similar polynomial functions. Let's embark on this mathematical journey to unravel the intricacies of function ranges.
Analyzing the Quartic Function y = -2x⁴ + 7
To determine the range of the quartic function y = -2x⁴ + 7, we need to analyze its structure and key components. A quartic function is a polynomial function of degree four, meaning the highest power of the variable x is 4. In our case, the function is in the form y = ax⁴ + c, where a = -2 and c = 7. The coefficient a plays a significant role in determining the function's overall shape and direction, while the constant term c influences its vertical shift. Understanding these elements is essential for accurately defining the range of the function.
The Role of the Leading Coefficient
The leading coefficient, which is -2 in our function, dictates the end behavior of the quartic function. Since the leading coefficient is negative, the graph of the function opens downwards. This means as x approaches positive or negative infinity, the value of y approaches negative infinity. The negative leading coefficient ensures that the function has a maximum value rather than a minimum. This characteristic is crucial in defining the upper bound of the range. The absolute value of the coefficient also affects the steepness of the graph; a larger absolute value results in a steeper curve.
Impact of the Constant Term
The constant term, which is 7 in our function, represents the vertical shift of the graph. It indicates the y-coordinate of the function's vertex (in this case, the maximum point). The graph of y = -2x⁴ is shifted upwards by 7 units to obtain the graph of y = -2x⁴ + 7. This vertical shift directly influences the upper limit of the range. Without the constant term, the maximum value would be 0, but with the addition of 7, the maximum value shifts to 7. This vertical translation is a fundamental aspect of understanding how constant terms affect the range of polynomial functions.
Determining the Range
Considering the downward-opening nature of the graph (due to the negative leading coefficient) and the vertical shift of 7 units, we can deduce the range of the function. The function has a maximum value at y = 7, which occurs when x = 0. Since the graph opens downwards, the function values will extend from 7 downwards to negative infinity. Therefore, the range of the function y = -2x⁴ + 7 includes all real numbers less than or equal to 7. This means y can take any value from negative infinity up to and including 7, but it will never exceed 7. This comprehensive analysis of the function's components allows us to definitively determine its range.
Step-by-Step Solution to Finding the Range
To find the range of the quartic function y = -2x⁴ + 7, we can follow a step-by-step approach that breaks down the problem into manageable parts. This method not only helps in understanding the solution but also provides a framework for solving similar problems. Each step focuses on a specific aspect of the function, ensuring a clear and logical progression towards the final answer. By systematically analyzing the function, we can confidently determine the set of all possible y-values.
Step 1: Analyze the Basic Quartic Term
The first step is to consider the basic quartic term, x⁴. The key characteristic of x⁴ is that it is always non-negative for any real number x. This is because raising any number to an even power results in a non-negative value. When x = 0, x⁴ = 0, and for any other value of x, x⁴ will be positive. This fundamental understanding is crucial as it forms the basis for analyzing the entire function. The non-negativity of x⁴ helps in determining the lower bound of the function's range before considering the effects of the leading coefficient and constant term.
Step 2: Consider the Leading Coefficient
Next, we consider the leading coefficient, which is -2 in y = -2x⁴ + 7. Multiplying x⁴ by -2 changes the direction of the graph. Since x⁴ is always non-negative, -2x⁴ will always be non-positive (i.e., negative or zero). This means the maximum value of -2x⁴ is 0, which occurs when x = 0. The negative leading coefficient flips the parabola vertically, causing it to open downwards. This step is essential for understanding the upper bound of the function before the vertical shift is applied. The leading coefficient's impact on the function's concavity significantly influences the range.
Step 3: Account for the Constant Term
Finally, we account for the constant term, which is +7 in y = -2x⁴ + 7. This term represents a vertical shift of the graph upwards by 7 units. The maximum value of -2x⁴ was 0, so adding 7 shifts this maximum value to 7. Thus, the function's maximum y-value is 7. Since the graph opens downwards, all other y-values will be less than or equal to 7. This vertical shift directly determines the upper limit of the range. The constant term acts as a crucial adjustment that finalizes the function's position on the coordinate plane.
Step 4: Determine the Range
Combining the above steps, we can determine the range of the function. The maximum y-value is 7, and since the graph opens downwards, the y-values extend to negative infinity. Therefore, the range of y = -2x⁴ + 7 is all real numbers y such that y ≤ 7. This range includes all values from negative infinity up to and including 7. Understanding each step in this process provides a clear and logical way to solve for the range of quartic functions and similar polynomial functions. By systematically analyzing each component, we arrive at a definitive answer for the range.
Why the Correct Answer is C: y ≤ 7
To understand why the correct answer is C, y ≤ 7, we need to revisit our analysis of the function y = -2x⁴ + 7. The process involves considering the effects of the quartic term, the leading coefficient, and the constant term. By breaking down the function into its core components, we can clearly see how the range is determined. This detailed explanation will solidify your understanding and demonstrate why the other options are incorrect.
The Impact of -2x⁴
The term -2x⁴ plays a crucial role in shaping the function's range. As discussed earlier, x⁴ is always non-negative for any real number x. When we multiply x⁴ by -2, the result, -2x⁴, becomes non-positive. This means that -2x⁴ can be zero or negative, but it will never be positive. The maximum value of -2x⁴ is 0, which occurs when x = 0. This understanding is fundamental because it sets the stage for how the constant term will ultimately define the range. The non-positive nature of this term is a key factor in determining the upper bound of the function.
The Effect of Adding 7
Adding 7 to -2x⁴ shifts the entire graph upwards by 7 units. This means the maximum value of the function, which was 0 for -2x⁴, is now 7. Therefore, the highest point the function reaches is y = 7. Since -2x⁴ is always non-positive, the function's values will never exceed 7. The constant term acts as a vertical translator, lifting the entire graph and setting the upper limit for the range. This upward shift is critical in defining the possible values of y.
Defining the Range: y ≤ 7
Given that the function's maximum value is 7 and the graph opens downwards due to the negative leading coefficient, the range of the function includes all real numbers less than or equal to 7. This is represented by the inequality y ≤ 7. The function values can be 7 (when x = 0) or any value less than 7, extending towards negative infinity. This range accurately captures the function's behavior, encompassing all possible y-values. Therefore, option C correctly describes the range of the function.
Why Other Options Are Incorrect
- Option A: y ≤ -2: This is incorrect because it only considers the leading coefficient and not the constant term. The constant term shifts the graph upwards by 7 units, making -2 an invalid upper bound.
- Option B: y ≥ 7: This is incorrect because the function opens downwards. The y-values will be less than or equal to 7, not greater than or equal to 7.
- Option D: All real numbers: This is incorrect because the function has a maximum value and does not extend to positive infinity. The negative leading coefficient limits the function's range to values less than or equal to 7.
By thoroughly analyzing the function and each of its components, we can confidently conclude that the correct answer is C: y ≤ 7. This explanation clarifies why this option is correct and why the others are not, reinforcing your understanding of how to determine the range of quartic functions.
Conclusion
In conclusion, determining the range of the quartic function y = -2x⁴ + 7 involves a comprehensive analysis of its components. The negative leading coefficient (-2) indicates that the graph opens downwards, while the constant term (+7) represents a vertical shift upwards by 7 units. This results in the function having a maximum value of 7 and extending to negative infinity. Therefore, the range of the function is y ≤ 7, which includes all real numbers less than or equal to 7.
Understanding the impact of the leading coefficient and constant term is crucial in determining the range of polynomial functions. The leading coefficient dictates the end behavior and direction of the graph, while the constant term shifts the graph vertically. By systematically analyzing these elements, we can accurately define the set of all possible y-values, which constitutes the range.
This article has provided a step-by-step approach to solving for the range, emphasizing the importance of breaking down the problem into manageable parts. By considering the quartic term, leading coefficient, and constant term, we can logically deduce the range of the function. This method not only helps in solving this specific problem but also provides a framework for analyzing similar polynomial functions. Through a clear understanding of these concepts, you can confidently tackle range-related problems in mathematics.
