Question

\(f(x)=-2(x-3)^2\)

Short answer

The given function \(f(x) = -2(x - 3)^2\) is a quadratic function. It is in standard form with \(a = -2\), \(h = 3\), and \(k = 0\). The vertex of the function is at the point \((3, 0)\). Since \(a < 0\), its graph will open downwards.

Step-by-step solution

Identify the standard form of the function

The given function is \(f(x) = -2(x - 3)^2\). This is a quadratic function in standard form \(f(x) = a(x-h)^2 + k\), where \(a\), \(h\), and \(k\) are constants, and \(a \neq 0\). For the given function, \(a = -2\), \(h = 3\), and \(k = 0\) (as it is not given in the equation).

Identify the vertex of the function

For a quadratic function \(f(x) = a(x-h)^2 + k\), the vertex is at the point \((h, k)\). For the given function, which is \(f(x) = -2(x - 3)^2\), the vertex is at the point \((3, 0)\).

Determine the direction of the parabola

When \(a < 0\), the graph of the quadratic function opens downwards. For the given function, \(f(x) = -2(x - 3)^2\), since \(a = -2\) and \(a < 0\), then the graph of the function opens downwards.

Key concepts

Standard Form of a Quadratic

Understanding the standard form of a quadratic function is critical for analyzing and graphing parabolas. The standard form is expressed as \(f(x) = a(x-h)^2 + k\), where \(a\), \(h\), and \(k\) are constants that determine the key characteristics of the parabola, and \(a \eq 0\).

The coefficient \(a\) affects the width and direction of the parabola. A larger absolute value of \(a\) means a 'narrower' parabola, while a smaller absolute value means a 'wider' parabola. The sign of \(a\) indicates the direction the parabola opens: a positive \(a\) opens upward, and a negative \(a\) opens downward.

The \(h\) and \(k\) values represent the coordinates for the vertex of the parabola, \(h\) affecting horizontal placement and \(k\) affecting vertical placement on the coordinate plane. To graph a quadratic function written in standard form, start by locating the vertex, then use the \(a\) value to determine the shape and direction of the parabola. By substituting different x-values into the function, you can plot additional points to sketch the graph's curve.

Vertex of a Parabola

The vertex of a parabola is one of the most significant features of its graph and represents the peak or the lowest point, depending on the direction in which the parabola opens. When dealing with the standard form \(f(x) = a(x-h)^2 + k\), finding the vertex is straightforward.

The coordinates of the vertex are given directly by \(h\) and \(k\), forming the ordered pair \( (h, k) \). This point is the maximum or minimum of the quadratic function. In the context of our example, \(f(x) = -2(x-3)^2\), the vertex is \( (3, 0) \), indicating that this is either the highest or lowest point on the graph based on whether the parabola opens upwards or downwards, respectively.

To visualize the vertex on a graph, plot the point and use it as a 'mirror' for the parabola’s symmetry. Each side of the parabola is a mirror image of the other, with the vertex being the point of symmetry. Hence, knowing the vertex helps in drawing an accurate graph of the quadratic function and understanding its properties.

Direction of a Parabola

The direction of a parabola is determined by the sign of the leading coefficient \(a\) in the quadratic function's standard form. This critical feature defines whether the parabola opens upwards or downwards, which has implications for the function's range and the maximum or minimum value it can have.

If \(a > 0\), the parabola opens upward, creating a 'U' shape. This means the vertex is at the lowest point, and as \(x\) moves away from the vertex, the value of \(f(x)\) increases. Conversely, if \(a < 0\), the parabola opens downward, like an upside-down 'U'. Here, the vertex is at the highest point, and \(f(x)\) decreases as \(x\) moves away from the vertex.

In our example, \(f(x) = -2(x-3)^2\), because the coefficient \(a = -2\) is negative, the parabola opens downwards. This means that the vertex represents the maximum value of the function. Understanding the direction of the parabola is essential for interpreting the graph and the behavior of the quadratic function, particularly when solving for optimization problems and analyzing projectile motion in physics.