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:
In this case, the standard form would be more helpful. This is because the standard form of a quadratic equation, ax^2 + bx + c = 0, explicitly shows the coefficients of the quadratic term, the linear term, and the constant term. These coefficients are necessary to factor the equation.
b. Graph the parabola:
In this case, the vertex form would be more helpful. The vertex form of a quadratic equation, f(x) = a(x − h)^2 + k, directly gives the coordinates of the vertex (h, k). This makes it easier to plot the vertex and sketch the shape of the parabola.
c. Identify the vertex, minimum, or maximum of the parabola:
Again, the vertex form would be more helpful here. The vertex form allows us to immediately read off the coordinates of the vertex, which are essential to identify the position of the vertex as well as the minimum or maximum value of the parabola.
d. Solve the equation using the quadratic formula:
In this case, the standard form would be more helpful. The quadratic formula is derived from the standard form of the quadratic equation. By plugging the coefficients from the standard form into the quadratic formula, we can easily find the solutions to the equation.