Let's start by considering the points (4,0) and (1,9). Since the y-coordinate of the first point is 0, we know that 4 is the x-intercept. It means that we want to write the equation of the parabola that passes through the point (1,9) and has x-intercept 4. To do so, we will use the intercept form of a quadratic function.
y=a(x-p)(x-q)
In this form, p and q are the intercepts. Therefore, we can partially write our equation.
y=a(x-4)(x- q)To obtain the quadratic equation we need at least three points.
Since our second given point (1,9) is not a x-intercept, we can choose any point at the x-axis to be our second x-intercept.
The only point that could not be chosen is (1,0), because the obtained relation would not be a function.
For simplicity, we will choose ( 0,0).
y=a(x-4)(x- 0) ⇔ y=ax(x-4)
Finally, since the parabola passes through the point (1,9), we can substitute 1 for x and 9 for y in our partial equation, and solve for a.
Knowing that a=- 3, we can write the full equation of the parabola.
y=- 3x(x-4)
Notice that this equation is written in intercept form. To write it in standard form we need to distribute - 3x.
