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When dealing with quadratic functions, we often need to determine their domain and range. The domain of a function is the set of all possible input values (usually represented by the variable \( x \)) that the function can accept. For quadratic functions like \( f(x) = 2x^2 + 6x - 7 \), the input can be any real number.
Therefore, the domain is all real numbers, which can be expressed as an interval: \((-fty, \infty)\). This essentially means there's no restriction on the values that \( x \) can take.
The range, on the other hand, is the set of all possible output values (usually represented by \( f(x) \)). The nature of a quadratic function as a parabolic function determines this. The graph of a parabola can either open upwards or downwards, depending on the sign of the coefficient of \( x^2 \).

• If the coefficient of \( x^2 \), \( a \), is positive, the parabola opens upwards.
• If \( a \) is negative, the parabola opens downwards.

In our function, since \( a = 2 \) (positive), the parabola opens upwards and consequently has a minimum point, the vertex. Hence, the range of our function is from the y-coordinate of the vertex to infinity: \([-rac{23}{2}, \infty)\).

Parabolic functions are a subset of polynomial functions of degree two. They are represented by the general form \( f(x) = ax^2 + bx + c \). These functions produce graphs known as parabolas, which are mirror-symmetrical and have distinct features such as a vertex, axis of symmetry, and either a maximum or minimum point.
Parabolas open upwards or downwards based on the coefficient of \( x^2 \). Here are some characteristics:

• If the parabola opens upwards (\( a > 0 \)), it has a minimum point.
• If it opens downwards (\( a < 0 \)), it has a maximum point.
• The vertex of the parabola represents either this minimum or maximum value.
• Parabolas are symmetric about a vertical line running through the vertex, called the axis of symmetry.

Understanding these basic properties helps predict the shape and behavior of a parabolic function graph.
The quadratic function \( f(x) = 2x^2 + 6x - 7 \) gives us an upward opening parabola due to the positive \( a \), indicating it has a minimum vertex. This important characteristic determines the function's range and behavior.

The vertex of a parabola is a crucial element in understanding the properties and graph of a quadratic function. It represents the point where the function reaches its minimum or maximum value. For the quadratic function \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex can be calculated using the formula \( x = -\frac{b}{2a} \).
For our function, \( a = 2 \) and \( b = 6 \), so substituting into the formula gives us:
\( x = -\frac{6}{2(2)} = -\frac{3}{2} \).
The y-coordinate of the vertex is found by plugging this x-value back into the original function. So,
\( f\left(-\frac{3}{2}\right) = 2\left(-\frac{3}{2}\right)^2 + 6\left(-\frac{3}{2}\right) - 7 \), which simplifies to \( -\frac{23}{2} \).
Thus, the vertex of \( f(x) = 2x^2 + 6x - 7 \) is \( \left(-\frac{3}{2}, -\frac{23}{2}\right) \). It defines the minimum point of the parabola, indicating the lowest output the function can achieve. This vertex plays a pivotal role in establishing the range and symmetry of the function.