Question

GUYS PLEASE I NEED HELP

Answer 1

Given:

The graphs of two parabolas.

To find:

The equation of the quadratic function with given graphs.

Solution:

(a)

The factor form of a parabola is

`y=a(x-b)(x-c)`

Where, a is a constant, b and c are x-intercepts.

From the graph (a) it is clear that the graph intersect the x-axis at -1 and 3. So, b=-1 and c=3.

`y=a(x-(-1))(x-3)`

`y=a(x+1)(x-3)` ...(i)

Put x=0 and y=3 because the y-intercept is (0,3).

`3=a(0+1)(0-3)`

`3=-3a`

`(3)/(-3)=a`

`-1=a`

Putting a=-1 in (i), we get

`y=-1(x+1)(x-3)`

`y=-(x+1)(x-3)`

Therefore, the equation of the parabola is
`y=-(x+1)(x-3)`.

(b)

The vertex form of a parabola is

`y=a(x-h)^2+k`

Where, a is a constant and (h,k) is vertex.

From the the graph (b), it is clear that the vertex of the of the parabola is at point (2,0) and y-axis is (0,8). So, h=2, k=0.

`y=a(x-2)^2+0`

`y=a(x-2)^2` ...(ii)

Put x=0 and y=8 because the y-intercept is (0,8).

`8=a(0-2)^2`

`8=4a`

`(8)/(4)=a`

`2=a`

Putting a=2 in (ii), we get

`y=2(x-2)^2`

Therefore, the equation of the parabola is
`y=2(x-2)^2`.