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 as listed below and explain why.
a. Factor the equation
b. Graph the parabola
c. Identify the vertex, minimum, and maximum of the parabola
d. Solve the equation using the quadratic formula
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a. Factor the equation:
In this case, the standard form of a quadratic equation would be more helpful because it allows us to easily identify the coefficients of the equation (a, b, and c), which are necessary for factoring. The standard form is given by: ax^2 + bx + c = 0.
b. Graph the parabola:
The vertex form of a quadratic equation would be more beneficial for this task. The vertex form is given by: f(x) = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex. Using the vertex form, it is easier to directly identify the vertex and determine the direction of the parabola's opening.
c. Identify the vertex, minimum, and maximum of the parabola:
Again, the vertex form would be more beneficial for this task. The vertex form explicitly gives the vertex coordinates, (h, k) and describes the shape of the parabola. By inspecting the coefficients a, h, and k in the vertex form equation, we can easily identify the vertex, the minimum (when a > 0), or the maximum (when a < 0) of the parabola.
d. Solve the equation using the quadratic formula:
The standard form of a quadratic equation is more helpful when solving the equation using the quadratic formula. The quadratic formula, x = (-b ± √(b^2 - 4ac))/(2a), requires the equation to be in standard form. By rearranging the equation in standard form (ax^2 + bx + c = 0), we can easily identify the coefficient values (a, b, and c) needed to substitute into the quadratic formula.
a. Factor the equation:
In order to factor a quadratic equation, the standard form is more beneficial than the vertex form. The standard form of a quadratic equation is given by: ax^2 + bx + c = 0. This form makes it easier to identify the coefficients (a, b, c) and use factoring techniques such as grouping, difference of squares, or perfect square trinomials. Factoring is typically easier when the equation is in standard form because it allows for a direct approach to finding the factors.
b. Graph the parabola:
The vertex form of a quadratic equation, which is given by: y = a(x - h)^2 + k, is more helpful for graphing the parabola. The vertex form provides the coordinates of the vertex, (h, k), which is essential for plotting the parabola on a graph. It also gives you information about the direction of the graph (upward or downward opening) based on the value of 'a'. By using the vertex form, you can easily determine the vertex and the axis of symmetry without any additional calculations.
c. Identify the vertex, minimum, and maximum of the parabola:
Similar to graphing, the vertex form is more beneficial for identifying the vertex, minimum, and maximum of a parabola. The vertex form directly provides the coordinates of the vertex (h, k). The vertex represents the minimum point if the parabola opens upwards or the maximum point if it opens downwards. By looking at the value of 'a', you can also determine whether it is a minimum or maximum. The vertex form simplifies the process of finding these key features without the need for additional calculations.
d. Solve the equation using the quadratic formula:
For solving the equation using the quadratic formula, the standard form of the quadratic equation is the most helpful. The quadratic formula is derived from the standard form, where x = (-b ± √(b^2 - 4ac)) / (2a). In order to substitute the coefficients (a, b, c) into the formula, it is easier to have the equation represented in the standard form. This way, you can directly substitute the coefficients and perform the necessary calculations to find the solutions for x.