91Ó°ÊÓ

The vertex of a quadratic function gives you a central point on its graph, usually forming the highest or lowest point on the curve. For a parabola, this is where the curve turns. Knowing the vertex is key because it helps to easily sketch the graph of the function.
To find the vertex of a quadratic function in the form of \(y = ax^2 + bx + c\), use the formula \(x = -\frac{b}{2a}\).
Once you have \(x\), substitute it back into the equation to find the \(y\)-value. For example, for the quadratic equation \(y = -2x^2 + 16x - 31\), we calculated the vertex at \((4, 1)\).
This reveals that the curve reaches its peak at this point, as the coefficient \(a\) is negative, indicating a parabola that opens downward.

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It always passes through the vertex of the parabola.
For the quadratic function \(y = ax^2 + bx + c\), the axis of symmetry is given by the formula \(x = -\frac{b}{2a}\), which means it shares the \(x\)-coordinate with the vertex.

• It's an important feature because it helps to predict the behavior of the graph.
• For example, any two points on opposite sides of the axis will be equidistant from it.

In the given quadratic \(y = -2x^2 + 16x - 31\), the axis of symmetry is at \(x = 4\), making it very simple to sketch the graph by reflecting points over this line.

The \(y\)-intercept of a quadratic function is the point where the parabola crosses the \(y\)-axis. This happens when \(x = 0\) in the equation. To find this intercept, simply set \(x = 0\) and solve for \(y\).
For the quadratic function \(y = -2x^2 + 16x - 31\), substituting \(x = 0\) yields \(y = -31\), meaning the \(y\)-intercept is \((0, -31)\).

• Knowing the \(y\)-intercept gives you a starting point for graphing the curve.
• It's essential for understanding where the parabola sits relative to the \(y\)-axis.

It provides a fixed point that helps confirm the parabola's directional trend and curve precision when graphing by hand.

A parabola is the symmetric curve formed by a quadratic function. It can open upwards or downwards depending on the sign of \(a\) in the equation \(y = ax^2 + bx + c\).
In our example, since \(a = -2\), the parabola opens downward.

• If \(a\) is positive, the parabola opens upward, resembling a cup.
• A negative \(a\) makes it open downward like an umbrella.

The vertex forms the turning point, and the axis of symmetry guarantees that one half of the parabola mirrors the other. Understanding these properties is useful for sketching and interpreting these graphs visually.

Graphing quadratic equations involves a few straightforward steps. Start by determining the vertex using the formula \(x = -\frac{b}{2a}\) and calculating the corresponding \(y\)-value.

• Find the axis of symmetry, which passes through the vertex vertically.
• Calculate the \(y\)-intercept by setting \(x = 0\).
• Recognize whether the parabola opens upwards or downwards based on the sign of \(a\).

Next, plot additional points to ensure accuracy, typically by choosing \(x\)-values close to the vertex. Finally, sketch the parabola by plotting all points, connecting them in a smooth curve reflecting across the axis of symmetry. This step-by-step approach helps visualize the beautiful symmetry and patterns of quadratic equations.