Note: Your teacher will grade your response to ensure you receive proper credit for your answer.
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 would be more helpful for factoring the equation. This is because the standard form, ax^2 + bx + c = 0, clearly shows the coefficients of each term (a, b, and c). By factoring the quadratic equation in standard form, we can easily find its roots by setting each factor equal to zero.
b. Graph the parabola:
The vertex form would be more helpful for graphing the parabola. The vertex form, a(x-h)^2 + k, provides direct information about the vertex of the parabola, which is represented by the values of h and k. By knowing the coordinates of the vertex, it becomes easier to plot the parabola accurately.
c. Identify the vertex, minimum, or maximum of the parabola:
The vertex form would be more helpful for identifying the vertex, minimum, or maximum of the parabola. In the vertex form, a(x-h)^2 + k, the values of h and k represent the coordinates of the vertex. The value of a determines whether the parabola opens upwards or downwards, indicating the maximum or minimum point respectively.
d. Solve the equation using the quadratic formula:
The standard form would be more helpful for solving the equation using the quadratic formula. This is because the quadratic formula, x = (-b ± √(b^2 - 4ac))/2a, directly makes use of the coefficients of the quadratic equation (a, b, and c). By substituting these values into the formula, we can find the solutions to the equation accurately.
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, and c, which are essential for factoring the equation. By factoring the quadratic equation, we can express it as a product of two binomials and find the solutions.
b. Graph the parabola:
The vertex form of a quadratic equation, f(x) = a(x-h)^2 + k, is more helpful when graphing the parabola. In vertex form, the equation provides the coordinates of the vertex (h,k) directly. By knowing the vertex, we can easily locate the minimum or maximum point on the graph. Moreover, the vertex form helps us identify whether the parabola opens upwards or downwards based on the sign of 'a'. This information aids in accurately sketching the complete parabola.
c. Identify the vertex, minimum, or maximum of the parabola:
Again, the vertex form of the quadratic equation is more helpful in identifying the vertex, minimum, or maximum of the parabola. In vertex form, the equation is given as f(x) = a(x-h)^2 + k. By analyzing the values of 'h' and 'k', we can determine the coordinates of the vertex, (h, k). Also, if 'a' is positive, the parabola opens upwards with a minimum at the vertex, and if 'a' is negative, the parabola opens downwards with a maximum at the vertex. Therefore, vertex form provides the necessary information to identify the vertex, minimum, or maximum easily.
d. Solve the equation using the quadratic formula:
For solving the quadratic equation using the quadratic formula, the standard form of the equation, ax^2 + bx + c = 0, is more helpful. The quadratic formula is derived from the standard form, where the coefficients a, b, and c can be directly substituted into the formula to obtain the solutions for 'x'. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a. Hence, having the equation in standard form facilitates the systematic substitution, simplification, and calculation required to find the solutions to the equation.
a. Factor the equation: The standard form would be more helpful when factoring the quadratic equation. This is because the standard form of a quadratic equation, which is in the form of ax^2 + bx + c = 0, allows us to easily identify the coefficients and determine the factors. By factoring, we can rewrite the equation in the form (x - r)(x - s) = 0, where r and s are the roots of the equation.
b. Graph the parabola: The vertex form would be more useful when graphing the parabola. The vertex form of a quadratic equation is given by y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. This form allows us to easily identify the vertex and the direction of the parabola's opening, making it easier to plot points and draw an accurate graph.
c. Identify the vertex, minimum, or maximum of the parabola: Again, the vertex form would be more beneficial when identifying the vertex, minimum, or maximum of the parabola. In vertex form, the vertex is explicitly given by the values of h and k. The coefficient 'a' also determines whether the parabola opens upward or downward. This form allows for easy identification of these key features.
d. Solve the equation using the quadratic formula: The standard form is more useful when solving the equation using the quadratic formula. The standard form ax^2 + bx + c = 0 allows us to easily identify the values of coefficients 'a', 'b', and 'c', which can then be plugged into the quadratic formula x = (-b ± sqrt(b^2 - 4ac)) / 2a for finding the solutions. Having the equation in standard form simplifies the process of applying the quadratic formula.