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The vertex of a parabola is a key point that helps us understand the shape and direction of the quadratic function's graph. For a quadratic equation in standard form, expressed as \( ax^2 + bx + c \), the vertex \(h, k\) can be found using a straightforward formula:

• \( h = -\frac{b}{2a} \)
• \( k \) is calculated by substituting \( h \) back into the function \( f(h) \)

Once you calculate \( h \), plug it into the function to get \( k \). This gives the vertex as a coordinate pair \( (h, k) \). For the quadratic \( f(x) = x^2 - x - 12 \), applying the formula results in the vertex \( (\frac{1}{2}, -\frac{47}{4}) \).
The vertex is not just a point on the graph; it represents the maximum or minimum value of the function, crucial for understanding how the parabola behaves and where it is located relative to the axes.

The x-intercepts of a parabola are the points where the graph cuts through the x-axis. These intercepts are crucial for sketching the graph accurately and understanding where the function has zero values. For quadratic functions, finding the x-intercepts involves solving the equation \( f(x) = 0 \).

For \( f(x) = x^2 - x - 12 \), you can factor the quadratic equation as \( (x - 4)(x + 3) = 0 \). Solving these factors reveals the x-intercepts:

• \( x = 4 \)
• \( x = -3 \)

These points, \((4, 0)\) and \((-3, 0)\), indicate where the parabola intersects the x-axis, providing reference points for drawing the curve and determining its intersections with horizontal lines.

The y-intercept is where the graph crosses the y-axis, representing the output of the function when the input \( x \) is zero. It is a straightforward calculation because it involves simply substituting zero into the function for \( x \).

For the quadratic \( f(x) = x^2 - x - 12 \), calculate the y-intercept by evaluating \( f(0) \):

• \( f(0) = 0^2 - 0 - 12 = -12 \)

Thus, the y-intercept is \((0, -12)\). Knowing the y-intercept helps locate another key point on the graph, which is especially handy for sketching a rough shape of the parabola and confirming its placement on the y-axis.

The direction in which a parabola opens is determined by the quadratic coefficient \( a \) in its standard form \( ax^2 + bx + c \). This direction tells us whether the function's values become larger (open upwards) or smaller (open downwards) as it moves away from the vertex.

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

For the function \( f(x) = x^2 - x - 12 \), the quadratic coefficient \( a = 1 \), which is positive, indicating that the parabola opens upwards.
This trait has a profound influence on the parabola's shape and means that the vertex is at the parabola's lowest point, making it a minimum of the function. Identifying the direction is crucial, as it plays a vital role in understanding the overall behavior of the function and the orientation of the graph.