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Let's start by understanding the quadratic equation given in the exercise: \(y = 3x^2 - 6\). This equation is a form of a quadratic function, as characterized by its highest exponent of 2. Quadratic equations typically take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In this case, \(a = 3\), \(b = 0\), and \(c = -6\). This tells us that the graph of this function will be a parabola. Quadratic equations are powerful tools in mathematics, often used to describe motion, area, and other physical phenomena. When \(a > 0\), the parabola opens upwards, and when \(a < 0\), it opens downwards. Here, because \(a = 3\) is positive, our parabola will open upwards.

Graphing a quadratic function involves plotting points that satisfy the equation and then connecting these points to form the characteristic parabolic curve. Start by selecting a few x-values, such as \(-2\), \(-1\), \(0\), \(1\), and \(2\). For each x-value, compute the corresponding y-value using the equation \(y = 3x^2 - 6\):

• When x = -2, \(y = 3(-2)^2 - 6 = 12 - 6 = 6\)
• When x = -1, \(y = 3(-1)^2 - 6 = 3 - 6 = -3\)
• When x = 0, \(y = 3(0)^2 - 6 = -6\)
• When x = 1, \(y = 3(1)^2 - 6 = 3 - 6 = -3\)
• When x = 2, \(y = 3(2)^2 - 6 = 12 - 6 = 6\)

Plot these points on a coordinate plane: (-2, 6), (-1, -3), (0, -6), (1, -3), and (2, 6). Connect the points with a smooth, U-shaped curve to show the parabola.

Parabolas have unique characteristics that provide valuable information about the quadratic function. Here are some key features:

• Vertex: This is the highest or lowest point on the parabola. In our function \(y = 3x^2 - 6\), the vertex is at (0, -6). Since our parabola opens upwards, this vertex represents the minimum value.
• Axis of symmetry: This is the vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. For our function, the axis of symmetry is the line \(x = 0\).
• Direction: As mentioned earlier, because the coefficient of \(x^2\) (which is 3) is positive, the parabola opens upwards.
• Intercepts: The y-intercept occurs where \(x = 0\), giving the point (0, -6). The function usually has two x-intercepts (roots), but our example does not cross the x-axis, showing imaginary roots.

Recognizing these characteristics in quadratic functions enables efficient graphing and better understanding of their behavior.

Analyzing the domain and range of a quadratic function is crucial.

Domain:
The domain includes all possible x-values for the function. For any quadratic function, you can input any real number into \(x\) and get a corresponding y-value. Thus, the domain here is all real numbers, written as \((-∞, +∞)\).

Range:
The range consists of all possible y-values the function can output. Since our function opens upwards and has its vertex at (0, -6), the smallest y-value is -6. As x moves further from zero in either direction, the y-values increase without bound. Therefore, the range is all real numbers y such that \(y \geq -6\). We write this as \textbf{\text{[-6, +∞)}} in interval notation.

Knowing the domain and range helps in understanding the scope and limitations of the function.