Question

Find the x-intercepts of the parabola with

vertex (-1,80) and y-intercept (0,75).

Write your answer in this form: (X1,Yz), (X2,72).

If necessary, round to the nearest hundredth.

Answer 1

the points (-5,0) and (3,0) are the x-intercepts of the parabola.

Explanation:

The equation of a parabola with vertex in (h,k) is:

`(x-h)^2=4p(y-k)`

Where p is a constant. Replacing the (h,k) by (-1,80), we get:

`(x+1)^2=4p(y-80)`

Additionally, we know that when x is 0, y is equal to 75. So, we can replace these values and solve for p as:

`(x+1)^2=4p(y-80)\\(0+1)^2=4p(75-80)\\1=4p(-5)\\p=(-1)/(20)`

So, replacing the value of p on the initial equation, we get that the equation of the parabola is equal to:

`(x+1)^2=4((-1)/(20))(y-80)\\(x+1)^2=(-1)/(5) (y-80)`

Solving for y, we get:

`x^2+2x+1=(-1)/(5)y+16 \\x^2+2x+1-16=(-1)/(5)y\\-5x^2-10x+75=y`

Then, the x-intercepts of the parabola are the values of x where y is equal to zero, so we need to solving the following equation:

`-5x^2-10x+75=0`

So, the values of x are calculated as:

`x_1=(10+√(10^2-4(-5)(75)) )/(2(-5))=-5\\x_1=(10-√(10^2-4(-5)(75)) )/(2(-5))=3\\`

Finally, the points (-5,0) and (3,0) are the x-intercepts of the parabola.