Question

How to create a graph of the polynomial function f(x) = ( x- a)(x - b)

Answer 1

Looking at the function f(x), it is in the factored form, so we can easily find the zeros of the function (that is, the x-intercepts) by equating each factor to zero:

`\begin{gathered} x-a=0\rightarrow x=a\\ \\ x-b=0\rightarrow x=b \end{gathered}`

The x-coordinate of the vertex is given by the average value of the zeros:

`\begin{gathered} x_v=(x_1+x_2)/(2)=(a+b)/(2)\\ \\ y_v=f(x_v)=((a+b)/(2)-a)((a+b)/(2)-b)=((-a+b)/(2))((a-b)/(2))=-((a-b)^2)/(4) \end{gathered}`

And the y-intercept can be found by using x = 0:

`\begin{gathered} f(0)=(0-a)(0-b)\\ \\ f(0)=ab \end{gathered}`

So the graph of this quadratic equation is given by: