Question

The image of a parabolic lens is projected onto a graph. The image crosses the x-axis at –2 and 3. The point (–1, 2) is also on the parabola. Which equation can be used to model the image of the lens?

y = (x – 2)(x + 3)
y = (x – 2)(x + 3)
y = (x + 2)(x – 3)
y = (x + 2)(x – 3)

Answer 1

Given:

The image of a lens crosses the x-axis at –2 and 3.

The point (–1, 2) is also on the parabola.

To find:

The equation that can be used to model the image of the lens.

Solution:

If the graph of polynomial intersect the x-axis at c, then (x-c) is a factor of the polynomial.

It is given that the image of a lens crosses the x-axis at –2 and 3. It means (x+2) and (x-3) are factors of the function.

So, the equation of the parabola is:

$y=k(x+2)(x-3)$          ...(i)

Where, k is a constant.

It is given that the point (–1, 2) is also on the parabola. It means the equation of the parabola must be satisfy by the point (-1,2).

Putting $x=-1, y=2$ in (i), we get

$2=k(-1+2)(-1-3)$

$2=k(1)(-4)$

$2=-4k$

Divide both sides by -4.

$\dfrac{2}{-4}=k$

$-\dfrac{1}{2}=k$

Putting $k=-\dfrac{1}{2}$ in (i), we get

$y=-\dfrac{1}{2}(x+2)(x-3)$

Therefore, the required equation of the parabola is $y=-\dfrac{1}{2}(x+2)(x-3)$.

Note: All options are incorrect.