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A quadratic function often appears in the form of \( y = ax^2 + bx + c \). The vertex of the parabola, a crucial point representing its peak or valley, can be determined using the **vertex formula**. This formula gives us the x-coordinate of the vertex: \( x = -\frac{b}{2a} \). In this formula, \(a\), \(b\), and \(c\) are coefficients from the quadratic equation.
Substitute the values of \(a = -1\) and \(b = 4\) from our example function \( y = -x^2 + 4x - 2 \).
The vertex x-coordinate is calculated as:
\[ x = -\frac{4}{2(-1)} = \frac{4}{2} = 2 \]
Once the x-coordinate is known, the y-coordinate can be found by substituting back into the function:
\[y = -(2)^2 + 4(2) - 2 = -4 + 8 - 2 = 2 \]
So, the vertex is located at the point \((2, 2)\). This point plays a key role in sketching the parabola, providing a powerful starting point for further analysis.

Once we have identified the vertex, graphing a parabola becomes more manageable.
The vertex, here at \((2, 2)\), is critical for understanding the parabola's behavior, such as its direction of opening. Because the coefficient of \(x^2\) (\(a = -1\)) is negative, this parabola opens downwards. Think of the vertex as the peak of a hill when \(a\) is negative.
Here's a step-by-step guide to graphing parabolas:

• Plot the vertex on the coordinate plane.
• The **axis of symmetry** assists in symmetry and runs vertically through the vertex. Our axis here is \(x = 2\).
• Choose x-values around the vertex to calculate additional points (e.g., \( x = 1, 3 \)). Find their respective y-values.
• Connect the points in a smooth curve to form the parabola.

Knowing these steps fosters a deeper understanding and visual representation of quadratic functions.

Quadratic functions are described by the coefficients \(a\), \(b\), and \(c\) in the expression \( y = ax^2 + bx + c \). These coefficients offer insights into the parabola's properties.
Here's what the coefficients mean:

• **Coefficient \(a\):** Determines the direction and width of the parabola.
  • If \(a > 0\), the parabola opens upwards.
  • If \(a < 0\), it opens downwards, like in our example, where \(a = -1\).
  • The greater the absolute value of \(a\), the narrower the parabola.
• **Coefficient \(b\):** Together with \(a\), impacts the x-value of the vertex. It influences the slope of the tangent line at the vertex.
• **Coefficient \(c\):** Represents the y-intercept, the point where the graph intersects the y-axis.

By understanding these coefficients, solving quadratic expressions for both analytical calculations and practical graphing becomes much clearer.