91Ó°ÊÓ

Understanding the vertex form of a parabola is critical when sketching its graph or analyzing its properties. The vertex form is written as:
\[y = a(x - h)^2 + k\]
In this equation,

• \(h\) and \(k\) represent the coordinates of the vertex, which is the highest or lowest point on the parabola, depending on whether it opens upwards or downwards.
• The variable \(a\) affects the direction and the width of the parabola. If \(a\) is positive, the parabola opens upward; if negative, it opens downward. Furthermore, the larger the absolute value of \(a\), the narrower the parabola.

The vertex form makes it straightforward to identify the vertex of the parabola. For example, a parabola with an equation \(y = 2(x - 3)^2 + 6\) has its vertex at the point \((3,6)\), which is easily read off from the equation. This form is particularly useful for sketching the graph as it allows the drafter to start at the vertex and work outwards to plot additional points.

The y-intercept of a parabola is the point where the graph crosses the y-axis. In algebraic terms, it's the value of \(y\) when \(x = 0\). This characteristic of a parabola is important because it provides a concrete point through which the curve will pass, aiding in the accurate sketching of its shape.
To find the y-intercept from the vertex form, simply set \(x = 0\) and solve for \(y\). For our example:
\[y = a(0 - 3)^2 + 6\]
The calculation will yield the y-intercept, which in the provided exercise is 2. This informs us that no matter what the value of \(a\) is, as long as we are sketching accurately, our parabola will cross the y-axis at the point \((0,2)\). Plotting both this point and the vertex allows one to start shaping the parabola on the graph.

The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola and divides it into two mirror-image halves. The equation of the axis of symmetry can always be found by using the \(h\) value of the vertex \((h, k)\). The formula for the axis of symmetry is:
\[x = h\]
This line is essential for sketching a parabola because it tells you that for every point on one side of the parabola, there is a corresponding point on the other side that is equidistant from the axis. In our example with the vertex at \((3,6)\), the axis of symmetry is the line \(x = 3\). So when you're drawing your parabola, after plotting the vertex and the y-intercept, ensure that each side of your graph is a mirrored reflection across the line \(x = 3\), which enforces the symmetry of the parabolic shape.