# Thread: Semi-circle function

1. ## Semi-circle function

The graph of y=(3x-x)^.5 is a semi-circle with diameter coincident with the positive x-axis, radius 1.5, center at (1.5,0). create a function passing through (3.5,5) and intersecting the x-axis at x=2 and x=5.

2. Hello, sjenkins!

The graph of y=(3x-x)^.5 is a semi-circle with diameter coincident
with the positive x-axis, radius 1.5, center at (1.5,0).

. . All this is all true . . . but who needs it ??

Create a function passing through (3.5, 5) and intersecting the x-axis at x=2 and x=5.
Make a sketch . . .
Code:
        |
|    (3½,5)
|       *
|
|
|
- - + - * - - - * - -
|   2       5
We can see that a parabola will work.

From the two x-intercepts, we have: . $f(x) \:=\:a(x-2)(x-5)$

To pass through $\left(\frac{7}{2},\:5\right)$, we have: . $a\left(\frac{7}{2}-2\right)\left(\frac{7}{2} - 5\right) \:=\:5$

. . $a\left(\frac{3}{2}\right)\left(-\frac{3}{2}\right) \:=\:5 \quad\Rightarrow\quad -\frac{9}{4}\,a \:=\:5 \quad\Rightarrow\quad a \:=\:-\frac{20}{9}$

Therefore: . $f(x) \:=\;-\frac{20}{9}(x-2)(x-5) \quad\Rightarrow\quad\boxed{ f(x) \;=\;-\frac{20}{9}x^2 + \frac{140}{9}x - \frac{200}{9}}$