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The vertex of a parabola is a crucial point that signifies its peak or trough. In mathematical terms, a parabola is described by a quadratic equation of the form \( y = ax^2 + bx + c \). The vertex of a parabola can be easily located using the vertex formula, \( x = -\frac{b}{2a} \). This formula allows us to find the x-coordinate of the vertex, which we then use to determine the y-coordinate by plugging it back into the original equation.

• For our example, with \( a = 2 \), \( b = -4 \), substituting into the formula gives \( x = 1 \).
• Next, substituting \( x = 1 \) back into the equation \( y = 2x^2 - 4x + 5 \), we find that \( y = 3 \).

Thus, the vertex is at the point \((1, 3)\). Understanding the vertex is important because it tells us about the parabolic path's highest or lowest point, crucial for graphing and real-world applications like physics and engineering.

The standard form of a quadratic equation is a necessary tool for understanding parabolas and their properties. It is expressed as \( y = ax^2 + bx + c \). This makes it straightforward to identify the equation's parameters and features, such as the parabola's width and orientation.

• Parameter \( a \): Determines the direction the parabola opens (upward if \( a > 0 \), downward if \( a < 0 \)) and its width (more narrow for larger \|a\| values).
• Parameters \( b \) and \( c \): Affect the parabola in terms of its horizontal shift and vertical shift on the y-axis.

In our given problem, \( a = 2 \), \( b = -4 \), and \( c = 5 \), resulting in a parabola which opens upwards. Each of these coefficients contributes to the curve's behavior and can guide you in creating its graph. This form is especially useful when you begin to locate other points on the parabola.

Converting a quadratic equation into vertex form can simplify the process of identifying the vertex and understanding the parabola's behavior. The vertex form is expressed as \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. This form provides immediate access to the vertex' position just by looking at the formula.

• \( a \) remains the same as in the standard form, indicating the direction and shape of the parabola.
• \( h \) and \( k \) are the x and y-coordinates of the vertex, making the equation easily interpretable.

For the equation \( y = 2(x-1)^2 + 3 \), we have explicitly derived the vertex as \((1, 3)\) just by examining the equation. This transformation simplifies both graphing and conceptualizing the curve's peak or lowest point. Transitioning from standard to vertex form is handy in many scenarios, particularly when you need to sketch the graph quickly or analyze the vertex's impact in real-world contexts.