91Ó°ÊÓ

A quadratic function of the form \( y = ax^2 + bx + c \) graphs as a parabola. The vertex is a very special point on this parabola. It's the maximum or minimum point, depending on whether the parabola opens upwards or downwards. In our specific exercise, the function is \( y = -3.1545 + 0.1914x - 0.0015x^2 \). Because the coefficient \( a = -0.0015 \) is negative, this parabola opens downwards.

The vertex of such a parabola gives us the peak point of the graph. To find the \( x \)-coordinate of the vertex, you can use the formula for quadratics, \( x = -\frac{b}{2a} \). Substituting \( a = -0.0015 \) and \( b = 0.1914 \) leads to \( x = -\frac{0.1914}{2 \times -0.0015} = 63.8 \). This calculation tells us that the vertex of this particular parabola occurs when \( x = 63.8 \).

To find out what the largest possible advertising intensity \( y \) will be, we need to calculate it at the vertex. Once we've determined the \( x \)-coordinate of the vertex (in this case, \( x = 63.8 \)), we substitute it into the original quadratic function to find \( y \).

Using the function \( y = -3.1545 + 0.1914x - 0.0015x^2 \), substitute \( x = 63.8 \):

• Calculate \( 0.1914 \times 63.8 = 12.21132 \)
• Calculate \( -0.0015 \times (63.8)^2 = -6.10392 \)
• Add it all together: \( y = -3.1545 + 12.21132 - 6.10392 \)

The resulting \( y \) value equals \( 2.95388 \), indicating the maximum advertising intensity at this level of concentration. When you solve such quadratic problems, the maximum or minimum value can always be found at the vertex of the parabola.

Graphing a quadratic function can reveal a lot of information visually, such as the vertex, intercepts, and the direction it opens. In the exercise, the quadratic \( y = -3.1545 + 0.1914x - 0.0015x^2 \), is a downward-opening parabola because the coefficient \( a = -0.0015 \) is negative.

Here are some important details to consider when graphing:

• The parabola reaches its peak at the vertex; calculated here as \( x = 63.8 \), \( y = 2.95388 \).
• The graph should be symmetrical about this vertex. Verifying this symmetry can ensure your graph is correct.
• For quick sketching, note that as \( x \) moves away from the vertex in both directions, the \( y \) value decreases, confirming the downwards opening.

The importance of graphing these functions lies in the visual confirmation it provides. By plotting, you can see that our calculated vertex represents the peak or maximum advertising intensity visually, which is a great aid in learning and understanding quadratics.