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Understanding the vertex of a parabola is essential when graphing quadratic equations. The vertex is the highest or lowest point on a parabola, depending on whether the parabola opens upwards or downwards. For the equation \(y=3x^2-2x+6\), we find the vertex by using the formula \( h = -\frac{b}{2a} \), where \(a\) and \(b\) are coefficients from the quadratic equation \(ax^2 + bx + c\).

In this case, \(a=3\) and \(b=-2\), giving us the x-coordinate of the vertex, \( x = \frac{1}{3} \). Substituting this back into the equation provides the y-coordinate of the vertex:\(y = 3(\frac{1}{3})^2 - 2(\frac{1}{3}) + 6\). By simplifying, we can identify the exact point of the vertex, which serves as a crucial reference for graphing the entire parabola.

Using the Vertex to Sketch the Parabola

Once the vertex is known, we can plot this point on the coordinate plane. It allows us to understand the direction in which the parabola opens and to draw its symmetrical shape accordingly. The vertex provides a clear starting point for sketching the curve of the parabola.

The axis of symmetry is a vertical line that divides a parabola into two mirror-image halves. Finding the axis of symmetry is a straightforward process once you've determined the vertex of the parabola. The equation of the axis of symmetry is always \(x = h\), where \(h\) is the x-coordinate of the vertex.

For the equation \(y=3x^2-2x+6\), as we've already established, the vertex is at \( h = \frac{1}{3} \), resulting in an axis of symmetry of \(x = \frac{1}{3}\). This line is essential for graphing because it tells us that for every point on one side of the parabola, there is an identical point mirrored across the line on the other side.

Importance of the Axis of Symmetry in Graphing

Graphically, the axis of symmetry helps ensure the proper shape and positioning of the parabola on the coordinate plane. It also aids in determining other points on the parabola by reflecting points across the axis to find their counterparts.

The y-intercept of a parabola is the point where it crosses the y-axis. This point is significant as it's one of the easiest to find and provides a reference for graphing the parabola. To find the y-intercept, we set \(x=0\) in the equation and solve for \(y\).

Looking at the specific equation \(y=3x^2-2x+6\), setting \(x=0\) gives us \(y = 3*0^2-2*0+6\). Simplifying, we see that the y-intercept is \(y=6\). This means that the point (0, 6) is where the parabola crosses the y-axis.

Simplifying Graphing with the Y-Intercept

The y-intercept is particularly useful in graphing because, along with the vertex and the axis of symmetry, it helps to anchor the parabola on the graph. Once it's plotted, you can use the symmetry of the parabola to find other points and further define the shape before sketching the curve of the parabola.