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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.

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) \.

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.

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.