91Ó°ÊÓ

The vertex of a parabola is a crucial point, as it represents the minimum or maximum value of the quadratic function. In the equation \( f(x) = \frac{2}{3}x^2 - 4 \), the polynomial is of the form \( ax^2 + bx + c \). For our function:

• \( a = \frac{2}{3} \)
• \( b = 0 \)
• \( c = -4 \)

The x-coordinate of the vertex is found using the formula \( x = -\frac{b}{2a} \). Since \( b = 0 \), it simplifies to \( x = 0 \). To find the y-coordinate, substitute \( x = 0 \) back into the function:

\[ f(0) = \frac{2}{3}(0)^2 - 4 = -4 \]

So, the vertex of our parabola is at \((0, -4)\). This point is where the parabola either reaches its lowest or highest value. Since \( a > 0 \), it's the lowest point and the parabola opens upwards.

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

Since the x-coordinate of the vertex is \( x = 0 \), the axis of symmetry for our function \( f(x) = \frac{2}{3} x^2 - 4 \) is the vertical line:

\[ x = 0 \]

This line goes through the vertex \( (0, -4) \) and helps in graphing the parabola symmetrically.

Understanding the domain and range of functions is essential for graphing.

Domain: For any quadratic function such as \( f(x) = ax^2 + bx + c \), the domain is all real numbers, because parabolas extend indefinitely along the x-axis. Hence, the domain of our function \( f(x) = \frac{2}{3} x^2 - 4 \) is:

\[ \text{Domain: } (-\infty, \infty) \]

Range: The range depends on the direction that the parabola opens. If \( a > 0 \), the parabola opens upwards, and the range starts from the y-coordinate of the vertex and goes to positive infinity. Since our \( a = \frac{2}{3} \) is positive, the parabola opens upwards, and the vertex is at \( (0, -4) \). Therefore, the range of our function is:

\[ \text{Range: } [-4, \, \infty) \]

This means the y-values start from -4 and go upwards to infinity.

A quadratic function is a second-degree polynomial and its graph is a parabola. The general form is \( f(x) = ax^2 + bx + c \), where:

• \( a \) determines the width and direction of the parabola (upwards if \( a > 0 \), downwards if \( a < 0 \))
• \( b \) affects the position of the vertex along the x-axis
• \( c \) is the y-intercept, where the graph crosses the y-axis

For the given function \( f(x) = \frac{2}{3}x^2 - 4 \):

• \( a = \frac{2}{3} \), so the parabola opens upwards and is relatively wide
• \( b = 0 \), placing the vertex along the y-axis
• \( c = -4 \), meaning the graph intersects the y-axis at -4

The quadratic function is central in various mathematical applications, ranging from physics to economics due to its predictable nature and symmetrical properties.