91Ó°ÊÓ

In quadratic functions, particularly those graphed as parabolas, the concept of a "relative maximum" is important. A relative maximum is the highest point in a particular region of a graph. If the parabola opens downwards, it will have a relative maximum at its vertex.
Why? Because all other points around it are lower. This point is where the graph reaches its peak before curves back down.
In our example exercise, the function given is \(f(x) = -x^2 + 3x - 2\), which forms a downward opening parabola. This is because the coefficient of \(x^2\) is negative.
Here, the vertex of the parabola is crucial because this is where the relative maximum occurs. In this case, at \(x = 1.5\), the function reaches its maximum value of \(-0.25\).

• **Downward opening parabola** — implies a relative maximum.
• **Vertex marks the relative maximum** — calculated using \(x = -b/2a\).
• **Relative maximum value is \(f(x)\) at the vertex**.

Graphing parabolas involves drawing the curve of a quadratic function. With any quadratic function \(ax^2 + bx + c\), the graph will take the shape of a parabola.
To properly graph such a function, follow these basics steps:

• **Determine the direction of the opening**: Whether the parabola opens upwards or downwards depends on the coefficient \(a\). If \(a>0\), it opens upwards. If \(a<0\), it opens downwards, like our example where \(a=-1\).
• **Identify the vertex of the parabola**: The vertex provides critical information on the parabola's turning point.
• **Plot the vertex and several points around it**: This helps in drawing a smoother curve.

Using a graphing utility makes visualizing these steps easier, especially for complex functions. In the example \(f(x) = -x^2 + 3x - 2\), plotting yields a clear downward sloping parabola that helps identify the behaviors and characteristics like the vertex and relative maximum.

The vertex of a parabola holds significant importance as it provides the peak (or trough) point of the graph depending on the direction of opening. In the quadratic function \(ax^2 + bx + c\), the vertex represents the point where the parabola changes direction.
For calculations, the vertex \((h, k)\) can be found using:

• **x-coordinate \(x = -b/2a\)**: Key formula derived from the standard form of the quadratic equation.
• **y-coordinate \(f(x)\)**: Substituting \(x\) back into the function gives the exact y value.

In our step-by-step solution, the vertex for the parabola \(f(x) = -x^2 + 3x - 2\) was calculated as \((1.5, -0.25)\). This tells us that at \(x = 1.5\), the maximum point on this downward opening parabola is \(-0.25\).
Understanding the vertex is vital as it not only defines the parabola's peak but also separates the graph into two symmetrical halves. This symmetry is a defining property of parabolas, making the vertex calculation crucial for graphing and analyzing quadratic functions.