Question

Describe the three forms of a quadratic function. Identify the vertex and axis if symmetry of each form. Which form do you prefer and why?

Answer 1

Explanation

The three forms of the quadratic equation can be given as

Part A

1) Standard Form

$StandardForm:f(x)=ax^2+bx+c$

2) Factored form

$FactoredForm:f(x)=a\left(x-p\right)\left(x-q\right)$

3) Vertex form

$VertexForm:f(x)=a\left(x-h\right)^2+k$

Part B

For the standard form, the vertex is given by

$Vertex=(-b/2a,f\left(-b/2a\right))$

For the factored form, the vertex is given by;

$Vertex=\left(\frac{p+q}{2},f(\frac{p+q}{2})\right)$

For the vertex form

$Vertex=(h,k)$

Part C

For the standard form, the axis of symmetry is given by

$x=-\frac{b}{2a}$

For the vertex form. the axis of symmetry is given by

$x=h$

For the factored form, the axis of symmetry is

$x=\frac{p+q}{2}$

Part C

I prefer the factored form because we can determine end behavior using the value of "a." Although the degree is not as easily identifiable, we know there are only two factors, making the degree two.

Also, we can easily identify the zeros or x-intercepts of the function. This is given by the values of p and q.

Lastly, we can easily get the vertex using the formula denoted above.

Only the factored form easily offers the three possiblities written above.