Find the Maximum/Minimum Value -360x^2-1440x+360
The problem involves a quadratic expression -360x^2 - 1440x + 360, where 'x' represents a variable. The task is to determine the maximum or minimum value of this expression. To identify this value, one must understand the nature of the quadratic curve represented by the expression, specifically whether it opens upwards or downwards, which is dictated by the sign of the coefficient of the x^2 term. Once the direction is established, the maximum or minimum value of the function can be found using methods from calculus or by completing the square to transform the expression into vertex form. The value will occur at the vertex point of the parabola.
$- 360 x^{2} - 1440 x + 360$
Answer
Solution:
Step 1:
The vertex of a parabola $y = ax^2 + bx + c$ is found at $x = - \frac{b}{2a}$. When $a < 0$, the parabola opens downward and the vertex represents the maximum point. Thus, the maximum value $y_{\text{max}}$ is obtained by evaluating the function at $x = - \frac{b}{2a}$.
Step 2:
Determine the vertex's $x$-coordinate using $x = - \frac{b}{2a}$.
Step 2.1:
Insert the given $a$ and $b$ values into the formula: $x = - \frac{-1440}{2(-360)}$.
Step 2.2:
Simplify the expression by eliminating the parentheses: $x = - \frac{-1440}{-720}$.
Step 2.3:
Further simplify the fraction.
Step 2.3.1:
Identify and cancel out common factors between the numerator and denominator.
Step 2.3.1.1:
Extract the factor of 2 from $-1440$: $x = - \frac{2 \cdot -720}{2 \cdot -360}$.
Step 2.3.1.2:
Eliminate the common factor of 2: $x = - \frac{\cancel{2} \cdot -720}{\cancel{2} \cdot -360}$.
Step 2.3.1.2.1:
Rewrite the simplified fraction: $x = - \frac{-720}{-360}$.
Step 2.3.1.2.2:
Reduce the fraction by dividing both the numerator and the denominator by $-360$: $x = - \frac{2}{1}$.
Step 2.3.1.2.3:
Complete the division to find the value of $x$: $x = -2$.
Step 3:
Compute the function's value at $x = -2$ to find the maximum value.
Step 3.1:
Substitute $x$ with $-2$ in the function: $f(-2) = -360(-2)^2 - 1440(-2) + 360$.
Step 3.2:
Carry out the simplification of each term.
Step 3.2.1:
Calculate the square of $-2$ and multiply by the coefficients: $f(-2) = -360 \cdot 4 + 1440 \cdot 2 + 360$.
Step 3.2.2:
Add the resulting values together: $f(-2) = -1440 + 2880 + 360$.
Step 3.2.3:
The maximum value of the function is $f(-2) = 1800$.
Step 4:
The maximum value of the function occurs at the point $(-2, 1800)$.
Knowledge Notes:
To find the maximum or minimum value of a quadratic function of the form $y = ax^2 + bx + c$, one can use the vertex formula. The vertex $(h, k)$ of the parabola is given by:
$h = - \frac{b}{2a}$, which is the $x$-coordinate of the vertex.
$k = f(h)$, which is the $y$-coordinate of the vertex and represents the maximum or minimum value of the function.
The sign of the coefficient $a$ determines whether the parabola opens upwards ($a > 0$) or downwards ($a < 0$), which in turn indicates whether the vertex is a minimum or maximum point, respectively.
When simplifying expressions, it is important to:
Cancel out common factors to reduce fractions.
Follow the order of operations, which is parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).
The maximum or minimum value of the quadratic function is significant in various applications, such as physics (projectile motion), economics (profit and cost analysis), and engineering (structural analysis).
