Question

Explain how you could write a quadratic function in factored form that would have a vertex with an x- coordinate of 3 and two distinct roots

Answer 1

the vertex form could be

(x - 3)^2 - c where c is the y coordinate of the vertex This graph would open upwards. The graph is symmetrical about the line x = 3

- so if it has 2 distinct roots they will be at equal intervals either side of x = 3 .

So we could make those root x = 1 and x = 5

f(x) = (x - 1 ) x - 5) would be an example of what you require

f(x) = (x - 8)(x + 2)

Answer 2

Answer with explanation:

It is given that vertex of Quadratic function has x coordinate equal to 3,and it has two distinct roots.

Suppose ,the vertex of Quadratic function ,which will be in the shape of parabola has coordinates ,(3,k).

→→Equation of Parabola ,having Vertex (3,k) ,can open in either of four directions that is along positive x-axis or negative x axis,or positive y-axis or negative y-axis.

So, any of the four equation are possible

1.→ y-b=(x-3)²

2. → y-c= - (x-3)²

3.→x-3=(y-m)²

4. → x-3= -(y-n)²

If you chose to find roots ,taking y as a independent variable,and x as a dependent variable, then you have to consider these two equations

3.→x-3=(y-m)²

4. → x-3= -(y-n)²

Such a two degree equation is

3. →x=(y-m)²+3

or,4. →x= -(y-n)²+3

→If you substitute ,the value of m by any real number in equation 3,it will not give real root.

So,this kind of Quadratic is not possible .

→→Now , in equation 4, if you will replace , n by any real number , and equate it with ,0 it will give two distinct real roots.

≡Now, Coming to Equation (1) and Equation (2)

1.→ y-b=(x-3)²

2. → y-c= - (x-3)²

here,if we consider , y as a dependent variable and x as an independent variable ,and then form Quadratic having x as a variable

1. y=(x-3)²+b

2. y= -(x-3)²+c

→In equation,1 if we limit the value of b from (-∞,0), and equate it with zero,there will be two real distinct roots.

⇒⇒In equation 2,if we limit the value of c from (0,∞),and equate it with zero,definitely there will be two distinct real roots.