A quadratic equation can be written in vertex form or in standard. 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 of the quadratic equation, which is in the form ax^2 + bx + c = 0, would be more helpful. This is because the standard form allows us to easily identify the coefficients a, b, and c, which are required to apply factoring methods such as factoring by grouping or using the difference of squares.
b. Graph the parabola:
The vertex form of the quadratic equation, which is in the form a(x - h)^2 + k, would be more helpful in graphing the parabola. The vertex form allows us to easily identify the coordinates of the vertex (h, k), which is essential in determining the shape and position of the parabola.
c. Identify the vertex, minimum, or maximum of the parabola:
Similar to graphing the parabola, the vertex form of the quadratic equation would be more beneficial in identifying the vertex, minimum, or maximum of the parabola. The vertex form directly provides the coordinates of the vertex (h, k) - the point where the parabola reaches its minimum or maximum value.
d. Solve the equation using the quadratic formula:
Here, the standard form of the quadratic equation would be more helpful. The quadratic formula is specifically derived from the standard form of the equation, and by substituting the coefficients a, b, and c into the formula, we can find the solutions for the quadratic equation.