Question

explain the benefits of each of the three forms of quadratic equations, standard form, vertex form, and factored form. What do they reveal about the graph or the quadratic?

Answer 1

Answer:

Summary:

Standard form allow us to quickly find the y-intercept.

Vertex form allow us to quickly locate the vertex.

And factored form allows us to quickly determine the roots/zeros.

Step-by-step explanation:

The three forms of quadratics are the standard form, vertex form, and the factored form. Each of them reveals a specific part about the quadratic.

Standard Form:

The standard form of a quadratic is:

$ax^2+bx+c$

There are only two details that can be conveyed by a quadratic in standard form immediately: (1) the leading coefficient a, and (2) the y-intercept.

The leading coefficient a will tell us if the parabola curves upwards or downwards.

And the constant c will give us the y-intercept.

Vertex Form:

The vertex form of a quadratic is:

$a(x-h)^2+k$

There are also two details that can be conveyed by a quadratic in vertex form:  (1) the vertex, and (2) the leading coefficient.

The leading coefficient is given by a. Again, this tells us the orientation of the parabola.

And the vertex is given by (h, k).

Hence, in my opinion, vertex form is the best form of a quadratic since it immediately reveals the vertex, the most important aspect of a quadratic.

Factored Form:

The factored form of a quadratic is:

$a(x-p)(x-q)$

Where p and q are the zeros/roots/solutions of the quadratic.

Again, factored form gives us two details about the quadratic: (1) the leading coefficient, and (2) the zeros.

The zeros tells us when the parabola crosses the x-axis, which can assist in graphing.

Summary:

Therefore, each form of a quadratic equation has its own benefits.

Standard form allow us to find the y-intercept.

Vertex form allow us to quickly locate the vertex.

And factored form allows us to quickly determine the roots/zeros.

Hence, depending on the question, each form can be useful in its own way.