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The vertex of a parabola is a key point that indicates the highest or lowest point on the graph of a quadratic function. In other words, it is the point where the graph changes direction. For a quadratic equation in the format \(y = ax^2 + bx + c\), the x-coordinate of the vertex can be found using the formula \(x = -\frac{b}{2a}\). This formula is derived from completing the square and helps identify the symmetry axis of the parabola.

• The value of \(x\) in the vertex provides the line of symmetry for the parabola.
• The y-coordinate is found by substituting the x-coordinate back into the original equation.

In our given equation \(y = x^2 - 6x + 5\), substituting \(a = 1\) and \(b = -6\) into the formula gives \(x = 3\). Substituting \(x = 3\) back into the equation, we find the corresponding y-value as \(-4\), giving the vertex \((3, -4)\). This point is crucial as it tells us that the parabola reaches its lowest point here, as the parabola opens upwards due to the positive \(a\) value.

The y-intercept of a quadratic function is where the graph crosses the y-axis. This occurs at the point where the input, or \(x\), is zero. To find this intercept, simply substitute \(x = 0\) into the quadratic equation and solve for \(y\).

• The y-intercept gives a direct point on the graph that is easy to plot.
• It also helps to confirm the shape and direction of the parabola when sketching by hand.

For our equation \(y = x^2 - 6x + 5\), substituting \(x = 0\) results in \(y = 5\), creating the point \((0, 5)\). By marking this point on the graph, it serves as a guide to ensure the parabola's curve is accurate and intersects the y-axis correctly. It supports the sketch, emphasizing the symmetry and the general shape influenced by the vertex.

Quadratic functions represent equations based on the format \(y = ax^2 + bx + c\). These functions generate a parabolic graph, offering several important characteristics like the vertex and y-intercept. Understanding these concepts is crucial for sketching and interpreting the graphs of quadratic functions.

• The coefficient \(a\) decides the parabola's opening direction (upward if positive, downward if negative).
• The vertex provides the turning point, allowing us to identify the function's maximum or minimum value.
• The y-intercept offers an easy-to-locate reference point.

Graphing quadratics involves using these features to sketch the curve. In the case of \(y = x^2 - 6x + 5\), the function is simplified by plotting known points like the vertex \((3, -4)\) and y-intercept \((0, 5)\). Align the symmetry of the parabola with these points to create an accurate representation. For more accuracy, a graphing calculator can validate the sketch by displaying the full parabola. This enhances understanding and ensures the graph accurately reflects the function's behavior.