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.
below are suppose the be the questions:
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
below are the answers:
Vertex form is most helpful for all of these tasks.
Let
.. f(x) = a(x -h) +k ... the function written in vertex form.
a) Factor:
.. (x -h +√(-k/a)) * (x -h -√(-k/a))
b) Graph:
.. It is a graph of y=x^2 with the vertex translated to (h, k) and vertically stretched by a factor of "a".
c) Vertex and Extreme:
.. The vertex is (h, k). It is a maximum if "a" is negative; a minimum otherwise.
d) Solutions:
.. The quadratic formula is based on the notion of completing the square. In vertex form, the square is already completed, so the roots are
.. x = h ± √(-k/a)
a. To factor the equation, the standard form of a quadratic equation would be more beneficial. In standard form, an equation is written as ax^2 + bx + c = 0, where a, b, and c are constants. This form allows us to easily identify the coefficients and therefore factorize the equation by finding the roots.
b. To graph the parabola, the vertex form of a quadratic equation would be more helpful. The vertex form is written as f(x) = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex. This form directly provides the coordinates of the vertex, making it easier to plot and sketch the parabola accurately.
c. To identify the vertex, minimum, or maximum of the parabola, the vertex form of a quadratic equation would be more beneficial. As mentioned earlier, the vertex form provides the coordinates of the vertex in a more straightforward manner. By looking at the equation in vertex form, you can instantly identify the vertex (h, k), whether it's at the lowest point (minimum) or the highest point (maximum) of the parabola.
d. To solve the equation using the quadratic formula, the standard form of a quadratic equation would be more helpful. The quadratic formula, x = (-b ± √(b^2 - 4ac)) / 2a, is derived based on the standard form of a quadratic equation. Substituting the coefficients from the standard form equation into the quadratic formula allows for a relatively straightforward mathematical calculation to find the solutions for x.
a. When factoring the quadratic equation, it is generally more beneficial to have it in standard form. The standard form of a quadratic equation is given as ax^2 + bx + c = 0, where a, b, and c are constants. This form allows us to easily identify the coefficients (a, b, and c) of the quadratic equation, which are important in the factoring process. By factoring, we can find the roots (or zeros) of the equation, which are the x-values that make the equation equal to zero.
b. When graphing the parabola, it is more helpful to have the quadratic equation in vertex form. The vertex form of a quadratic equation is given as a(x - h)^2 + k, where (h, k) represents the vertex of the parabola, and a determines the direction and scale of the parabola. Having the equation in vertex form allows us to easily identify the vertex (h, k) of the parabola, which allows us to locate the turning point. It also helps us determine if the parabola opens upwards or downwards.
c. To identify the vertex, minimum, or maximum of the parabola, it is most helpful to have the equation in vertex form. As mentioned earlier, the vertex form of a quadratic equation is given as a(x - h)^2 + k. By identifying the values of h and k, we can directly determine the coordinates of the vertex (h, k). The vertex represents the minimum or maximum point on the parabola, and knowing its coordinates can help determine the direction and shape of the parabola.
d. When solving the equation using the quadratic formula, it is most helpful to have the equation in standard form. The quadratic formula is given as x = (-b ± √(b^2 - 4ac))/2a, which allows us to find the roots of the quadratic equation. By having the equation in standard form, we can easily identify the required coefficients (a, b, and c) needed for the quadratic formula. Plugging these values into the formula allows us to solve for the roots (or zeros) of the equation.
A. Standard
B. Vertex
C. Vertex
D. Standard
a) clearly in standard form
b) I find the vertex form useful, since you immediately have one point and the axis of symmetry. All you need is another point and its reflection in the axis of symmetry and you should be able to sketch the graph
c) If you have to find the vertex, max/mins etc, what form contains the word vertex ??
d) The formula is defined according to a, b, and c of
ax^2 + bx + c = 0 . So what do you think?