Question

Find the x-intercepts of the parabola with

vertex (4,75) and y-intercept (0,27).
Write your answer in this form: (X1,Y1), (X2,72).
If necessary, round to the nearest hundredth.

Answer 1

The x intercepts are (-1, 0) and (9, 0).

Explanation:

We can write the equation in vertex form:

y = a(x - b)^2 + c

Here b = 4 and c = 75 so we have

y = a(x - 4)^2 + 75 where a is a constant to be found.

The y-intercept is (0,27) so

27 = a(0 - 4)^2 + 75

16a = 27 - 75

a = -48/16 = -3

So to find the x-intercepts we solve the equation:

-3(x - 4)^2 + 75 = 0

(x - 4)^2 = -75 / -3 = 25

x - 4 = +/- √25

x = 5+ 4 = 9 , -5 + 4 = -1.

Answer 2

(- 1, 0), (9, 0)

Explanation:

The equation of a parabola in vertex form is

y = a(x - h)² + k

where (h, k) are the coordinates of the vertex and a is a multiplier

here (h, k) = (4, 75), so

y = a(x - 4)² + 75

To find a substitute (0, 27) into the equation

27 = a(- 4)² + 75 , that is

27 = 16a + 75 ( subtract 75 from both sides )

16a = - 48 ( divide both sides by 16 )

a = - 3, thus

y = - 3(x - 4)² + 75 ← equation in vertex form

To obtain the x- intercepts let y = 0

- 3(x - 4)² + 75 = 0 ( subtract 75 from both sides )

- 3(x - 4)² = - 75 ( divide both sides by - 3 )

(x - 4)² = 25 ( take the square root of both sides )

x - 4 = ±
`√(25)` = ± 5

Add 4 to both sides

x = 4 ± 5, hence

x = 4 - 5 = - 1 or x = 4 + 5 = 9

Substitute these values into the equation for corresponding values of y

x = - 1 : y = - 3(- 5)² + 75 = - 75 + 75 = 0 → (- 1, 0)

x = 9 : y = - 3(5)² + 75 = = 75 + 75 = 0 → (9, 0)

The x- intercepts are (- 1, 0), (9, 0)