A quadratic equation can be written in vertex form or in standard form. Sometimes one form is more beneficial than the other.
Identify which form would be more helpful if you needed to do each task listed below and explain why.
a. Factor the equation.
b. Graph the parabola.
c. Identify the vertex, minimum, or maximum of the parabola.
d. Solve the equation using the quadratic formula.
a. Factor the equation:
The standard form would be more helpful if you needed to factor the equation. This is because the standard form of a quadratic equation, ax^2 + bx + c = 0, explicitly shows the coefficients of each term, making it easier to identify common factors or patterns that can be used to factor the equation.
b. Graph the parabola:
The vertex form would be more helpful if you needed to graph the parabola. This is because the vertex form of a quadratic equation, a(x-h)^2 + k, directly provides the coordinates of the vertex (h, k), making it easier to plot the vertex and determine the shape and direction of the parabola.
c. Identify the vertex, minimum, or maximum of the parabola:
Again, the vertex form would be more helpful in identifying the vertex, minimum, or maximum of the parabola. As mentioned earlier, the vertex form provides the coordinates of the vertex directly, making it easier to determine its position on the graph and whether it is a minimum or maximum point.
d. Solve the equation using the quadratic formula:
Both forms, vertex form or standard form, can be used to solve the equation using the quadratic formula. However, the standard form might be slightly more convenient since it explicitly provides the coefficients of each term, making it easier to substitute into the quadratic formula. Additionally, the standard form allows for a more straightforward comparison to the quadratic formula, which is traditionally written in terms of a, b, and c in the standard form.