Constraints For Quadratic Functions Ensuring A Minimum Value Of 600

To effectively determine a reasonable constraint for the given quadratic function, ${ y = -2x^2 + 40x + 600 }$, ensuring it remains at or above 600, a detailed analysis is essential. This involves understanding the properties of quadratic functions, particularly how their coefficients affect their shape and behavior. Additionally, it's crucial to consider the practical implications of different constraints on ${ x }$, which influences the function's output.

Understanding the Quadratic Function

At its core, our function is a quadratic equation represented as ${ y = -2x^2 + 40x + 600 }$. The coefficient of the ${ x^2 }$ term, which is -2, indicates that the parabola opens downward. This is a crucial piece of information because it tells us that the function has a maximum value. Finding this maximum value and where it occurs is pivotal in setting our constraints. The other coefficients, 40 for the ${ x }$ term and 600 as the constant term, influence the parabola's position and vertical shift on the graph.

To find the vertex of the parabola, which represents the maximum point, we use the formula ${ x = -b / (2a) }$, where ${ a }$ and ${ b }$ are the coefficients of the ${ x^2 }$ and ${ x }$ terms, respectively. Plugging in our values, ${ a = -2 }$ and ${ b = 40 }$, we get:

${ x = -40 / (2 * -2) = 10 }$

This tells us that the maximum value of the function occurs when ${ x = 10 }$. To find the actual maximum value (the ${ y }$-coordinate of the vertex), we substitute ${ x = 10 }$ back into the original equation:

${ y = -2(10)^2 + 40(10) + 600 = -200 + 400 + 600 = 800 }$

So, the vertex of the parabola is at the point ${ (10, 800) }$, confirming that the maximum value of the function is 800. Understanding this vertex is key because it helps us visualize the function's behavior and determine the range of ${ x }$ values for which the function will be at or above 600.

Graphical Analysis and Implications

Visualizing the graph of this quadratic function is incredibly helpful. Imagine a parabola opening downwards, with its peak at the point ${ (10, 800) }$. The function starts at ${ y = 600 }$ when ${ x = 0 }$, climbs to its peak at ${ y = 800 }$ when ${ x = 10 }$, and then descends again. The critical points where the function's value is exactly 600 are of particular interest to us.

To find these points, we set the function equal to 600 and solve for ${ x }$:

${ 600 = -2x^2 + 40x + 600 }$

Subtracting 600 from both sides simplifies the equation:

${ 0 = -2x^2 + 40x }$

We can factor out a ${ -2x }$ from the equation:

${ 0 = -2x(x - 20) }$

This gives us two solutions for ${ x }$: ${ x = 0 }$ and ${ x = 20 }$. These ${ x }$-values are where the parabola intersects the line ${ y = 600 }$. This means that the function's value is at or above 600 for all ${ x }$ values between 0 and 20, inclusive.

Practical Constraints and Considerations

Understanding the graphical behavior of the function allows us to evaluate the given constraint options effectively. We know that the function is at least 600 within the interval ${ 0 \leq x \leq 20 }$. This interval is crucial for defining a reasonable constraint.

Considering the provided options:

• ${ 0 \leq x \leq 20 }$: This is a valid constraint. As we determined through our calculations, the function's value is at or above 600 within this interval.
• ${ -10 \leq x \leq 30 }$: While this interval includes the region where the function is above 600, it also includes values of ${ x }$ where the function is less than 600. For example, if ${ x = -10 }$, the function's value would be significantly lower than 600. Therefore, this constraint is not suitable.
• ${ x \geq 0 }$: This constraint is too broad. Although the function starts at 600 when ${ x = 0 }$, it eventually decreases below 600 as ${ x }$ increases beyond 20. Thus, this is not a reasonable constraint.
• All real numbers: This is the least restrictive constraint and, consequently, the least suitable. The function dips below 600 for ${ x }$ values outside the ${ [0, 20] }$ interval, making this an inappropriate constraint.

Evaluating Constraint Options

In this section, we critically assess each constraint option to pinpoint the one that reliably maintains the quadratic function's value at or above 600. This involves both analytical evaluation and an understanding of the function's behavior across different intervals.

Option 1: ${ 0 \leq x \leq 20 }$

As previously calculated, the quadratic function ${ y = -2x^2 + 40x + 600 }$ intersects the line ${ y = 600 }$ at ${ x = 0 }$ and ${ x = 20 }$. Within this interval, the parabola is at or above the ${ y = 600 }$ level, reaching its maximum value of 800 at ${ x = 10 }$. Therefore, this constraint ensures the function meets the minimum value requirement and is a viable option.

Option 2: ${ -10 \leq x \leq 30 }$

This constraint expands the interval significantly beyond the region where the function is at least 600. To understand why this is problematic, consider values outside the ${ [0, 20] }$ range. For instance, let's evaluate the function at ${ x = -10 }$:

${ y = -2(-10)^2 + 40(-10) + 600 = -200 - 400 + 600 = 0 }$

At ${ x = -10 }$, the function's value is 0, which is far below 600. Similarly, for values of ${ x }$ greater than 20, the function's value will also fall below 600 due to the downward-opening nature of the parabola. Thus, this constraint is too broad and includes regions where the function does not meet the required condition.

Option 3: ${ x \geq 0 }$

This constraint sets a lower bound for ${ x }$ but doesn't provide an upper bound. We know that the function is at or above 600 between ${ x = 0 }$ and ${ x = 20 }$. However, beyond ${ x = 20 }$, the function starts decreasing. Let's test a value greater than 20, say ${ x = 25 }$:

${ y = -2(25)^2 + 40(25) + 600 = -1250 + 1000 + 600 = 350 }$

At ${ x = 25 }$, the function's value is 350, which is significantly less than 600. This demonstrates that the constraint ${ x \geq 0 }$ is insufficient because it doesn't restrict ${ x }$ values where the function falls below the required threshold.

Option 4: All real numbers

This option is the most unrestricted and, therefore, the least likely to be suitable. We've already established that the function dips below 600 for ${ x }$ values outside the ${ [0, 20] }$ interval. Including all real numbers would mean incorporating a vast range of ${ x }$ values for which the function's value is well below 600. Thus, this constraint is not viable.

Conclusion: Determining the Optimal Constraint

After thoroughly analyzing each constraint option in the context of the quadratic function ${ y = -2x^2 + 40x + 600 }$, the most reasonable constraint to ensure the function is at least 600 is ${ 0 \leq x \leq 20 }$. This constraint accurately reflects the interval within which the function's value meets or exceeds the required threshold. Other options either include regions where the function's value falls below 600 or are too restrictive, failing to capture the entire range where the function satisfies the condition.

In summary, understanding the graphical representation of the quadratic function, calculating the vertex, and determining the points of intersection with the line ${ y = 600 }$ are essential steps in identifying the correct constraint. This approach ensures that the chosen constraint aligns perfectly with the function's behavior and the stipulated minimum value requirement.