Problem 60 - Algebra for College Students

Question

Find the maximum or minimum value of \(y\) for each function. $$y=-3 x^{2}+24 x$$

Step-by-Step Solution

Answer

The maximum value of \(y\) is 48.

Step 1: Identify the given function

The given function is a quadratic equation: \(y = -3x^2 + 24x\).

Step 2: Determine the form of the quadratic equation

The standard form of a quadratic equation is \(ax^2 + bx + c\). Comparing this with \(y = -3x^2 + 24x\), we note that \(a = -3\), \(b = 24\), and \(c = 0\).

Step 3: Identify the coefficients

Here, \(a = -3\) and \(b = 24\).

Step 4: Determine the vertex of the parabola

The vertex form of a quadratic function \(y = ax^2 + bx + c\) can be found using \(x = -\frac{b}{2a}\). Substituting the values of \(a\) and \(b\), we get \(x = -\frac{24}{2(-3)} = 4\).

Step 5: Calculate the corresponding y-coordinate

Substitute \(x = 4\) back into the original function: \(y = -3(4)^2 + 24(4) = -3 \cdot 16 + 96 = -48 + 96 = 48\).

Step 6: Identify the maximum or minimum value

Since the quadratic term has a negative coefficient (\(a = -3\)), the parabola opens downward. Thus, the vertex represents the maximum value.

Step 7: State the maximum value

The maximum value of \(y\) is 48.

Key Concepts

quadratic equation

A quadratic equation is a type of polynomial equation of the form:
\[ ax^2 + bx + c = 0 \]
Here, a, b, and c are constants, and x is the variable. The highest degree of the variable x in a quadratic equation is 2, which gives it the 'quadratic' name.
In our example, the quadratic equation is given by:
\[ y = -3x^2 + 24x \]
This means a = -3, b = 24, and c = 0.

Quadratic equations form a parabola when graphed. The direction in which the parabola opens (upwards or downwards) is determined by the sign of the coefficient a.

• If a is positive, the parabola opens upwards and has a minimum point.
• If a is negative, the parabola opens downwards and has a maximum point.

vertex form

The vertex form of a quadratic function is an alternative way to express a quadratic equation. It focuses on the vertex of the parabola, which is either a maximum or minimum point.

The vertex form is given by:
\[ y = a(x - h)^2 + k \]
Here, the vertex of the parabola is at the point \(h, k\). To convert the standard form \( ax^2 + bx + c \) to the vertex form, we use a process involving completing the square.

In our example problem, we find the vertex by using the formula:
\[ x = -\frac{b}{2a} \]
Plugging in the coefficients \(a = -3\text{ and }b = 24\), we get:
\[ x = -\frac{24}{2(-3)} = 4 \]

This means the x-coordinate of the vertex is 4. We then substitute x back into the original equation to find the corresponding y-coordinate:
\[ y = -3(4)^2 + 24(4) = 48 \]
Therefore, the vertex is at the point \ (4, 48) \.

maximum value

In a quadratic function, the vertex of the parabola can either represent a maximum or minimum value.
Since the coefficient a in our example is negative (\( a = -3 \)), the parabola opens downwards. This tells us the vertex represents the maximum value of the function.

To find the maximum value:

• First, determine the x-coordinate of the vertex: \( x = -\frac{b}{2a} = 4 \)
• Secondly, substitute x back into the quadratic function to find the y-coordinate:
  \( y = -3(4)^2 + 24(4) = 48 \)

The maximum value of the function is thus 48. This means that the highest point on the parabola (and thus the highest value that y can reach) is 48.

coefficients

In a quadratic equation, the coefficients are the numbers in front of the variables. They determine the shape and position of the parabola.
- a is the coefficient of \(x^2\) and affects the width and direction (up or down) of the parabola.
- b is the coefficient of \(x\), and it influences the position horizontally.
- c is the constant term and represents the y-intercept, or where the parabola crosses the y-axis.

In our specific example:

• The coefficient a is -3, indicating the parabola opens downwards and is relatively narrow.
• The coefficient b is 24, which helps to place the vertex horizontally at \(x = 4\).
• The constant c is 0, meaning the parabola crosses the y-axis at the origin (0,0).

Understanding the role of each coefficient allows us to predict and describe the graph of the quadratic function accurately.