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Quadratic functions are mathematical expressions of the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. The graph of a quadratic function is called a parabola. This parabola can either open upwards or downwards depending on the sign of the coefficient \(a\).

• If \(a > 0\), the parabola opens upwards, like a U-shape.
• If \(a < 0\), the parabola opens downwards, like an inverted U.

Understanding the basics of quadratic functions is essential for solving many types of problems, including finding the vertex, determining the vertex form, identifying the range of the function, and understanding where the function increases or decreases.

The vertex form of a quadratic function provides an alternative way to express the function, highlighting the vertex's coordinates directly. A quadratic function in vertex form is given by:\[ f(x) = a(x-h)^2 + k \]In this form, \( (h, k) \) represents the vertex of the parabola. Converting a function from standard form to vertex form often involves completing the square. For the given function\[ f(x) = 2x^2 + 4x - 16 \],we find its vertex using the formulas: \[ h = -\frac{b}{2a} \]and\[ k = f(h) \].In our case, we have \(h = -1\) and \(k = -18\), so the vertex form of the function is\[ f(x) = 2(x + 1)^2 - 18 \].Identifying the vertex is crucial for understanding the function's properties.

The minimum value of a quadratic function is the y-coordinate of the vertex when the parabola opens upwards (i.e., \(a > 0\)). For our function\[ f(x) = 2x^2 + 4x - 16 \],we calculated the vertex to be at \( (-1, -18) \). Since the coefficient \(a = 2\) is positive, the function opens upwards and has a minimum value at the vertex. Therefore, the minimum value of the function is \[ k = -18 \]. This minimum value is the lowest point on the graph of the function.

The range of a quadratic function is the set of all possible y-values. To determine the range, we look at the direction in which the parabola opens. For our function\[ f(x) = 2x^2 + 4x - 16 \],which opens upwards (since \(a = 2 > 0\)), the y-values start from the minimum value and extend to positive infinity. The vertex provides us with this minimum y-value, which is \( -18 \). So, the range of the function is\[ [-18, \, \infty) \].This means the function takes all values from \(-18\) upwards.

To determine where the quadratic function is increasing or decreasing, we look at the vertex and the direction the parabola opens. For the function\[ f(x) = 2x^2 + 4x - 16 \],the vertex is at \((-1, -18)\), and since the parabola opens upwards, it decreases to the left of the vertex and increases to the right of it.

• Decreasing Interval: \((-\infty, -1)\)
• Increasing Interval: \((-1, \infty)\)

In simpler terms, the function gets smaller as \(x\) moves from \(-\infty\) to \(-1\), and gets larger as \(x\) moves from \(-1\) to \(+\infty\). Understanding these intervals helps where the function rises and falls.