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The vertex of a quadratic function is a crucial point on its graph. It's the highest or lowest point, depending on the direction the parabola opens. When the quadratic function is in the standard form \( ax^2 + bx +c \), you determine the vertex using the formula \( x = -\frac{b}{2a} \).

For the function \( f(x) = x^2 - 4 \), the coefficients are \( a = 1 \), \( b = 0 \), and \( c = -4 \). Plug these values into the formula:

• \( x = -\frac{0}{2 \times 1} = 0 \)

To find the \( y \)-coordinate of the vertex, substitute \( x = 0 \) back into the equation to get \( f(0) = 0^2 - 4 = -4 \).

This means the vertex is located at \((0, -4)\). This point is essential for sketching the graph.

The direction in which a parabola opens is determined by the coefficient \( a \) in the standard form \( ax^2 + bx + c \). If \( a > 0 \), the parabola opens upward. If \( a < 0 \), it opens downward.

In our quadratic function, \( f(x) = x^2 - 4 \), we have \( a = 1 \), which is a positive number.

• This tells us that the parabola opens upward.

This direction means the graph will have a minimum point at the vertex and will extend infinitely upwards. Knowing the direction is necessary when graphing the function, ensuring that the shape accurately reflects the equation. By identifying whether a parabola is opening upwards or downwards, you can more easily spot maximum or minimum value scenarios in a real-world context.

To graph a quadratic function completely, you'll need to determine where it intersects the axes: the intercepts.
**Calculating the \( y \)-intercept:**

• To find the \( y \)-intercept, set \( x = 0 \) in the function. For \( f(x) = x^2 - 4 \), this is simply \( f(0) = 0^2 - 4 = -4 \).
• The \( y \)-intercept is at the point \((0, -4)\).

**Calculating the \( x \)-intercepts:**

• Set the function equal to zero: \( x^2 - 4 = 0 \).
• Factorize the equation: \((x - 2)(x + 2) = 0\).
• Solving gives \( x = 2 \) and \( x = -2 \).
• The \( x \)-intercepts are the points \((2, 0)\) and \((-2, 0)\).

Each intercept is a point where the graph intersects an axis, providing crucial reference points for sketching the parabola.

Sketching the graph of a quadratic function involves plotting key points and drawing the correct shape of the curve. Start by marking the vertex on the coordinate plane. For \( f(x) = x^2 - 4 \), plot the vertex at \((0, -4)\).

Then, add the \( x \)-intercepts at \((2, 0)\) and \((-2, 0)\), and the \( y \)-intercept again at \((0, -4)\). These points give a framework for the graph.

• Connect these points with a smooth, curved line reflecting the direction instructed by \( a \).
• In this case, ensure the parabola opens upwards from the vertex.

The resulting curve should pass through all plotted points, expanding outward as it moves away from the vertex. This visual representation provides a clear picture of how the function behaves across different inputs. Remember, drawing by hand or in a digital tool takes some practice, but with these steps, you can accurately depict the function's graph.