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 of a quadratic equation (ax^2 + bx + c = 0) is more helpful when factoring the equation. This is because the standard form allows us to easily identify the coefficients (a, b, c), which are needed for factoring. Factoring the equation involves finding two binomials that, when multiplied, will result in the original quadratic equation. The standard form provides a clear representation of the coefficients, making it easier to determine the factors.
b. Graph the parabola:
The vertex form of a quadratic equation (a(x - h)^2 + k) is more helpful when graphing the parabola. This form explicitly provides the coordinates of the vertex (h, k), which is crucial for accurately sketching the parabola on a graph. By simply reading the values of h and k, we can determine the exact location of the vertex, allowing us to easily plot and sketch the parabola.
c. Identify the vertex, minimum, or maximum of the parabola:
Again, the vertex form of a quadratic equation is more helpful when identifying the vertex, minimum, or maximum of the parabola. In vertex form, the equation is presented as a transformation of the basic quadratic function y = x^2. The values of h and k in the vertex form directly reveal the coordinates of the vertex (h, k), which is essential for determining the location of the minimum or maximum point of the parabola.
d. Solve the equation using the quadratic formula:
For solving the equation using the quadratic formula, the standard form of a quadratic equation is more helpful. The quadratic formula is derived from the standard form, where the coefficients a, b, and c can be easily identified. By substituting these values into the quadratic formula (x = (-b ± √(b^2 - 4ac)) / (2a)), we can solve for the values of x that satisfy the equation. The standard form simplifies the process of plugging in the coefficients into the quadratic formula, making it more convenient for solving the equation.