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Parabolas are U-shaped graphs that can open either upwards or downwards. They are the graphs of quadratic functions, which are equations of the form \(y = ax^2 + bx + c\). Here's what you need to know about parabolas:

When the coefficient \(a\) is positive, the parabola opens upwards, resembling a happy face. When \(a\) is negative, it opens downwards, resembling a sad face.

The point at which the parabola bends, or the 'tip' of the U, is called the vertex. The vertex can either be a maximum (if the parabola opens downward) or a minimum (if it opens upward). Understanding parabolas is crucial for solving quadratic function problems, such as finding maximum and minimum values. Let's break down these concepts further.

The maximum and minimum values of a quadratic function depend on the vertex of the parabola. Here's what you need to know:

• If the parabola opens upwards (\(a > 0\)), the vertex represents the minimum point.
• If the parabola opens downwards (\(a < 0\)), the vertex is the maximum point.

For the function \(y = -9x^2 + 16\), the coefficient \(a\) is -9, which is negative. This tells us the parabola opens downwards. Therefore, the vertex will give us the maximum value.

To find the maximum or minimum value, substitute the x-coordinate of the vertex back into the original equation. The result will be the highest or lowest y-value the function can have.

Vertex form makes it easier to find the vertex of a parabola directly. The vertex form of a quadratic function is:

\[y = a(x - h)^2 + k\]

where \(h\) and \(k\) represent the coordinates (x, y) of the vertex. However, if our function is in standard form \((y = ax^2 + bx + c)\), we can still find the vertex:

• The x-coordinate of the vertex is found using \(-b/2a\).
• If there is no x-term (\(b = 0\), then the vertex's x-coordinate is simply 0.

In our example, \(y = -9x^2 + 16\), there is no x-term, so the vertex is at \(x = 0\). Substituting this back into the equation, we get the maximum value \(y = 16\). Hence, the vertex form helps us quickly identify the highest or lowest point on the graph, making it a valuable tool in analyzing quadratic functions.