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The vertex of a parabola is the point where the curve changes direction. In the standard form of a quadratic function, which is \( f(x) = ax^2 + bx + c \), the vertex can be calculated using the formula \( x = -\frac{b}{2a} \). This gives the x-coordinate of the vertex. To find the y-coordinate, simply substitute this x-value back into the original equation.

For the given equation \( f(x) = 3x^2 \), where \( a = 3 \), \( b = 0 \), and \( c = 0 \), the x-coordinate of the vertex is:
\( x = -\frac{0}{2 \times 3} = 0 \).
Substituting \( x = 0 \) back into the function, we get:
\( f(0) = 3(0)^2 = 0 \).

Therefore, the vertex is (0, 0). This point is crucial as it represents the minimum point of the parabola since the coefficient of \( x^2 \) is positive.

The axis of symmetry of a parabola is a vertical line that divides the parabola into two mirror-image halves. For a quadratic function in the form \( f(x) = ax^2 + bx + c \), the axis of symmetry can be found using the same formula used to find the x-coordinate of the vertex: \( x = -\frac{b}{2a} \).

In our example, the equation is \( f(x) = 3x^2 \), which gives us:
\( x = -\frac{0}{2 \times 3} = 0 \).

This means the axis of symmetry is the vertical line \( x = 0 \). Understanding the axis of symmetry helps in graphing the parabola accurately, as the parabola will be symmetric around this line.

A quadratic function is a type of polynomial function of degree two. It is represented in the standard form as \( f(x) = ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). The graph of a quadratic function is a U-shaped curve known as a parabola.

Key properties of a quadratic function include:

• It has a vertex, which is either the highest or lowest point of the parabola.
• It has an axis of symmetry, which is a vertical line that passes through the vertex.
• If \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.

In the equation \( f(x) = 3x^2 \), the function simplifies to \( ax^2 \). Here, the coefficient \( a \) is 3, indicating that the parabola opens upwards.

The domain and range of a function tell us the possible values that the function can take.

The domain of a quadratic function is all real numbers, denoted as \( (-\infty, \infty) \). This is because we can substitute any real number for \( x \) within the function.

The range of a quadratic function depends on whether the parabola opens upwards or downwards:

• If the parabola opens upwards (\( a > 0 \)), the range is \([y_{min}, \infty) \).
• If the parabola opens downwards (\( a < 0 \)), the range is \((-\infty, y_{max}] \).

For the function \( f(x) = 3x^2 \), the parabola opens upwards since \( a = 3 \). The vertex at (0, 0) is the minimum point. Therefore, the range is all real numbers greater than or equal to 0, which is denoted as \( [0, \infty) \). Understanding the domain and range is essential for graphing and analyzing the behavior of the function.