Write an equation of the parabola in intercept form.
(3,4) (4,0) (2,0)
Nope, not possible
the robot tutor would have a root of (3,0).
since (4,0) and (2,0) are x-intercepts, they must be roots.
so
y = a(x - 4)(x - 2)
with (3,4) on ...
4 = a(-1)(1)
a = -4
y = -4(x-4)(x-2)
or
y = -4x^2 + 24x - 32
since the roots are at 2 and 4, the vertex must lie on the line x=3.
Since the point (3,4) lies on the parabola, it must be the vertex, so
y = a(x-3)^2 + 4
since +2 is a root,
a+4 = 0, so
y = -4(x-3)^2 + 4
which agrees with the solution above
To write the equation of a parabola in intercept form, we need to find the x-intercepts and the y-intercept of the parabola.
Given the three points (3,4), (4,0), and (2,0), we can observe that the parabola intersects the x-axis at x = 4 and x = 2. Hence, the x-intercepts are 4 and 2.
The y-intercept is the point where the parabola intersects the y-axis. From the three given points, we can see that the parabola intersects the y-axis at y = 4.
Using this information, we can write the equation of the parabola in intercept form as follows:
(x - x1)(x - x2) = y - y-intercept
where x1 and x2 are the x-intercepts, and y-intercept is the y-coordinate of the y-intercept point.
Substituting the values into the equation, we get:
(x - 4)(x - 2) = y - 4
Expanding the equation, we have:
(x^2 - 6x + 8) = y - 4
Rearranging the equation to the standard form, we get:
x^2 - 6x + 8 - y + 4 = 0
Simplifying further, we have:
x^2 - 6x - y + 12 = 0
Therefore, the equation of the parabola in intercept form is:
x^2 - 6x - y + 12 = 0