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.
Here's how each form of the quadratic equation is helpful in different scenarios:
a. Factor the equation:
Standard Form is more beneficial for factoring. The Standard Form of a quadratic equation is written as ax^2 + bx + c = 0. This form is directly suited to factorization methods (assuming the quadratic is factorable) where you look for two binomials that multiply to give ax^2 + bx + c.
b. Graph the parabola:
Both forms can be helpful, but the Vertex Form offers a more immediate visualization for graphing a parabola. The Vertex Form of a quadratic equation is written as y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. Since you have the vertex, you can easily plot it on a graph and sketch the parabola using the direction of the opening (upwards for a > 0 and downwards for a < 0) and the magnitude of a to determine the width.
c. Identify the vertex, minimum, or maximum of the parabola:
Vertex Form is ideal for quickly identifying the vertex of a parabola, as well as its minimum or maximum value. As stated earlier, the vertex is (h, k) in the equation y = a(x - h)^2 + k. If a is positive, the vertex is a minimum; if a is negative, the vertex is a maximum. This direct readout of the information makes it the preferred form for this task.
d. Solve the equation using the quadratic formula:
Standard Form is suited for directly applying the quadratic formula. The quadratic formula, x = [-b ± sqrt(b^2 - 4ac)]/(2a), is derived from the Standard Form ax^2 + bx + c = 0. It allows you to find the solutions (roots) of the equation by substituting the coefficients a, b, and c into the formula. You don't need to manipulate the equation further to use the quadratic formula if the equation is already in Standard Form.
Each form provides specific advantages depending on the information you need or the task you're performing with the quadratic equation.