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

Answer:

answer below

Step-by-step explanation:

The vertex lies on the axis of symmetry, so the axis of symmetry is x = 3. Find any two x-intercepts that are equal distance from the axis of symmetry. Use those  x-intercepts to write factors of the function by subtracting their values from x. For example, 2 and 4 are each 1 unit from x = 3, so f(x) =  (x – 2)(x – 4) is a possible function.

Answer 2

The definition of quadratic function is given by:

$(1) \ f(x)=ax^{2}+bx+c \\ \\ where \ a, b, c \ are \ real \ values$

So the problem asks for two conditions:

1. Write the quadratic function in factored form.
2. The vertex has an x-coordinate of 3.

Then the equation (1) must be written as follows:

$f(x)=(x-m)(x-n) \\ \\ where \ m \ and \ n \ are \ the \ roots$

To satisfy the two conditions:

$\frac{m+n}{2}=3$

So we are free to choose the value of one root, say:

$m=1$

Thus:

$n=6-1=5$

Finally, the answer is:

$f(x)=(x-1)(x-5)$

The graph of this function is shown in the figure below.