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The vertex of a parabola is a critical point that emphasizes its highest or lowest position on a coordinate plane. For quadratic functions in the form of \( ax^2 + bx + c \), the vertex can be identified using the formula \( x = -\frac{b}{2a} \). This equation helps to find the x-coordinate of the vertex. After finding the x-value, you can compute the corresponding y-coordinate by substituting back into the original function.

In the example quadratic function \( f(x) = x^2 - 4 \), the coefficient \( b = 0 \) makes the calculation straightforward. Plugging the values into the formula gives \( x = -\frac{0}{2 \times 1} = 0 \). Then, substituting \( x = 0 \) into the function, we find \( f(0) = 0^2 - 4 = -4 \). Thus, the vertex of this parabola is at \( (0, -4) \).

The vertex tells us important information about the parabola. If \( a > 0 \), it is the lowest point (a minimum), and if \( a < 0 \), it's the highest point (a maximum). For \( f(x) = x^2 - 4 \), the vertex \( (0, -4) \) is a minimum.

Intercepts are the specific points where a function intersects the x-axis and y-axis.

**Y-Intercept:** This is where the graph crosses the y-axis, occurring when \( x = 0 \). For the function \( f(x) = x^2 - 4 \), the y-intercept is found by substituting \( x = 0 \) into the equation, resulting in \( f(0) = -4 \). Therefore, the y-intercept is at \( (0, -4) \).

**X-Intercepts:** X-intercepts are found by setting the function equal to zero and solving for \( x \). So, for the equation \( x^2 - 4 = 0 \), solve for \( x \) by setting it to zero:

• First, rearrange to \( x^2 = 4 \).
• Then, take the square root, giving \( x = \pm 2 \).

This yields x-intercepts at \( (2, 0) \) and \( (-2, 0) \).

Intercepts are fundamental for sketching quadratic functions, providing crucial markers that define the graph's path across the axes.

Graphing a quadratic equation, such as \( f(x) = x^2 - 4 \), involves plotting points that shape the function into a parabola.

To begin:

• Identify the vertex previously found (\( (0, -4) \) for our example).
• Mark the intercepts: y-intercept at \( (0, -4) \) and x-intercepts at \( (2, 0) \) and \( (-2, 0) \).

Since the coefficient \( a = 1 \) is greater than zero, the parabola opens upwards, providing symmetry around its axis, usually the y-axis.

Sketching involves:

• Starting from the vertex, draw a smooth curve passing through the intercept points.
• Ensure the parabola is symmetrical, meaning both sides mirror each other across the y-axis.

Graphing helps to visually understand the nature of quadratic functions, offering a picture of how they behave when variables change.