Assessment and Learning in Knowledge Spaces (ALEKS) Basic Math Placement Practice Test

How do you find the vertex of a parabola given in standard form?

1. By using the equation x = -b/2a

2. By calculating y = ax² + bx + c

3. By using the formula h = -b/(2a)

4. By determining the roots of the equation

The correct answer is: By using the equation x = -b/2a

To find the vertex of a parabola represented in standard form, which is \( y = ax^2 + bx + c \), the key lies in understanding the relationship between the coefficients \( a \) and \( b \) and the vertex's x-coordinate. The formula \( x = -\frac{b}{2a} \) is derived from completing the square or using calculus methods to find the minimum or maximum point of the quadratic function. This formula specifically gives the x-coordinate, denoted as \( h \), of the vertex of the parabola. Since a parabola opens either upward or downward depending on the sign of \( a \), determining the vertex allows us to understand the behavior and location of the parabola better. Once the x-coordinate \( h \) is found using \( -\frac{b}{2a} \), you can substitute this value back into the original equation \( y = ax^2 + bx + c \) to find the corresponding y-coordinate \( k \), which completes the vertex \((h, k)\). Thus, using this x-coordinate formula is a fundamental first step to locating the vertex accurately. The other options, while related to quadratic functions, do not directly lead to the vertex