Question

Find the equation in factored form y = a(x-r1)(x-r2) using the following points:

(9,0) (2,5) (-1,0)
(-1,-3) (-5, 0) (1, 0)
a. y = -5(x-2)(x+1)
b. y = 5(x-9)(x+1)
c. y = -5(x+9)(x+1)
d. y = 5(x-9)(x-2)

Answer 1

To derive the equation in factored form, we identify the zeros of the function from the points where y=0 and then determine the leading coefficient 'a' by using a point where y is not zero. The correct equation from the given options that fits all the points is y = 5(x-9)(x+1).

Step-by-step explanation:

To find the equation in factored form y = a(x-r1)(x-r2) using the given points, we can follow these steps:

1. Use the points where the y-coordinate is zero to identify the roots (r1 and r2) of the equation. This gives us (x-9), (x+1), and (x+5) because the x-values of the points where y = 0 are 9, -1, and -5, respectively.
2. Since the polynomial crosses the x-axis at x = 1, we also have (x-1) as a factor.
3. Plug in the point (2,5) to solve for the leading coefficient 'a'. Using y = a(x-9)(x+1)(x+5)(x-1), we substitute x=2 and y=5:
4. Calculate the value of 'a' from the equation above.
5. The correct equation that fits all the given points would be the one with the roots and 'a' that we've just calculated.

From the options provided, we can see that the factored form needs to include the terms (x-9) and (x+1) because of the x-intercepts at x=9 and x=-1. We also need to include (x+5) and (x-1) as factors. Notably, one of the options provided, option b, matches these requirements:

y = 5(x-9)(x+1)