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The vertex of a parabola is a crucial point. For a quadratic function of the form \( f(x) = ax^2 + bx + c \), the vertex provides the location of either the minimum or maximum on its graph. To find the vertex, use the formula \( x = -\frac{b}{2a} \). This expression gives you the x-coordinate of the vertex. After determining the x-coordinate, plug it back into the function to find the corresponding y-coordinate.
In our example \( f(x) = x^2 - 4 \), we see that \( b = 0 \). Therefore, the x-coordinate of the vertex is \( 0 \). Substituting \( x = 0 \) into \( f(x) \) yields \( f(0) = -4 \), making the vertex (0, -4). This point signifies the lowest position in this specific upward opening parabola.

Intercepts are where the graph crosses the axes. They provide additional points to guide the accurate sketching of a quadratic function.

• Y-intercept: Found by evaluating the function at \( x = 0 \), giving the point where the graph intersects the y-axis. In this case, substituting \( x = 0 \) in \( f(x) = x^2 - 4 \) gives \( f(0) = -4 \), so the y-intercept is at (0, -4).
• X-intercepts: Found by setting the function equal to zero. Solving \( x^2 - 4 = 0 \) involves finding when the function crosses the x-axis. This gives us \( x^2 = 4 \), leading to \( x = \pm 2 \). Therefore, x-intercepts occur at (2, 0) and (-2, 0).

Intercepts play a significant role in shaping the graph, providing reference points along axes.

Graphing a quadratic equation involves plotting key points on the coordinate plane. It includes placing the vertex, intercepts, and drawing the curve that represents the equation.
To sketch \( f(x) = x^2 - 4 \), begin by marking the vertex at (0, -4). It's also the y-intercept here. Next, place the x-intercepts, (2, 0) and (-2, 0), on the graph.
Connect these points with a smooth, continuous curve, ensuring that it opens upwards according to the direction established by the \( a \) value. These steps help construct a parabola that accurately depicts the quadratic equation.

The direction a parabola opens is determined by the sign of the coefficient \( a \) in the quadratic expression \( ax^2 + bx + c \).

• If \( a > 0 \), the parabola opens upwards, resembling a U-shape. Its vertex is the lowest point, as in our example \( f(x) = x^2 - 4 \) where \( a = 1 \).
• If \( a < 0 \), the parabola opens downwards, forming an upside-down U, and the vertex becomes the highest point.

This direction is vital, as it influences where the vertex lies concerning the rest of the graph. In practical terms, it affects whether we interpret the vertex as a maximum or a minimum point.