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In the world of quadratic functions, the vertex is a crucial concept. It represents the point on the graph where the curve either reaches its peak or its lowest point. To find the vertex of a quadratic function in the form \( ax^2 + bx + c \), we use the formula \( x = -\frac{b}{2a} \). This formula helps us locate the x-coordinate of the vertex. After finding the x-coordinate, substitute it back into the function to find the corresponding y-coordinate. For the example function \( f(x) = x^2 + x - 6 \), using \( a = 1 \) and \( b = 1 \), we calculated \( x = -\frac{1}{2} \). Plugging this back into the equation gives us the y-coordinate \( -\frac{25}{4} \). Thus, the vertex of our parabola is \( \left(-\frac{1}{2}, -\frac{25}{4}\right) \). This point is key in sketching the graph and understanding its shape.

The direction of opening for a parabola is dictated by the coefficient \( a \) in the quadratic function \( ax^2 + bx + c \). This direction tells you if the parabola points up or down. It's simple:

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

For the function \( f(x) = x^2 + x - 6 \), since \( a = 1 \) which is greater than zero, the parabola opens upwards. Recognizing the direction of opening is essential for graph sketching because it helps visualize how the parabola behaves and where it positions its vertex.

Intercepts of a quadratic function are the points where the graph crosses the axes.

Y-Intercept:

This is where the graph crosses the y-axis, occurring where \( x = 0 \). Simply substitute 0 into the function: \( f(0) = 0^2 + 0 - 6 = -6 \). Thus, the y-intercept is \( (0, -6) \).

X-Intercepts:

These are the points where the graph crosses the x-axis, where \( f(x) = 0 \). For our function, solve the equation: \( x^2 + x - 6 = 0 \) by factoring \((x - 2)(x + 3) = 0\). The solutions to this equation, \( x = 2 \) and \( x = -3 \), are the x-intercepts at points \( (2, 0) \) and \( (-3, 0) \). Knowing intercepts allow you to pinpoint exactly where the parabola meets the axes, providing crucial anchors in the graph.

Sketching a quadratic graph involves piecing together all the information we've gathered. Start by plotting the vertex \( (-\frac{1}{2}, -\frac{25}{4}) \). This point will be the center of symmetry for your parabola. Next, plot the y-intercept \( (0, -6) \) and the x-intercepts \( (2, 0) \) and \( (-3, 0) \). With these points marked, you can draw the parabola, making sure it's symmetrical around the vertex. Because the parabola opens upwards due to our positive \( a \), shape the curve with a gentle rise on both sides from the vertex. Visualizing the graph as a smooth curve helps you understand the function's behavior and how it changes with varying x-values.