91Ó°ÊÓ

The vertex form of a quadratic equation is a helpful way to easily identify the vertex of the parabola. The general form of a quadratic equation is given by \(y = ax^2 + bx + c\). However, the vertex form rewrites it as \(y = a(x - h)^2 + k\), where \(h\) and \(k\) represent the coordinates of the vertex (\(h, k\)).

To convert a quadratic equation from standard form to vertex form, you can use the method of completing the square. Here's a simplified process:

• Start with the quadratic equation in standard form \(y = ax^2 + bx + c\)
• Factor out the coefficient \(a\) from the first two terms if \(a eq 1\)
• Rewrite the quadratic and linear terms in perfect square form by completing the square
• Add and subtract the same value inside the squared term
• Simplify the equation to vertex form

Using this form makes it easy to identify the vertex and sketch the graph of the quadratic function. Vertex form tells us whether the parabola opens upwards or downwards (based on the sign of \(a\)) and where its highest or lowest point is located.

The y-intercept of a quadratic function is the point where the graph intersects the y-axis. To find this, simply set \(x = 0\) in the quadratic equation and solve for \(y\).

For example, consider the quadratic equation \(y = x^2 + x - 12\). By setting \(x = 0\):
\(y = 0^2 + 0 - 12 = -12\). Thus, the y-intercept is at the point \(0, -12\).

Finding the y-intercept gives you a key point that helps sketch the graph of the quadratic function. It represents the starting point where the parabola crosses the y-axis, making it a reliable anchor for visualizing the entire curve. Knowing the y-intercept, along with the vertex, allows for a more accurate sketch of the parabola.

In coordinate geometry, a parabola is the graph of a quadratic function. It takes a distinct U-shape and can open either upwards or downwards.

Important characteristics of a parabola include:

• **Vertex**: The highest or lowest point on the parabola, where it changes direction
• **Axis of Symmetry**: A vertical line that runs through the vertex and divides the parabola into two mirror-image halves
• **Direction of Opening**: Determined by the sign of the coefficient of \(x^2\) (positive signifies upwards, negative signifies downwards)
• **Y-Intercept**: The point where the parabola crosses the y-axis

For instance, the quadratic function \(y = 2x^2 - 12x - 20\) describes an upward-opening parabola because its leading coefficient (2) is positive.

Understanding these elements helps in accurately sketching and analyzing quadratic functions. The vertex provides the key point from which the symmetry and shape of the parabola can be drawn, while the y-intercept offers an additional point of reference on the graph.