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
[tex]y=a(x-b)(x-c)[/tex]
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.
[tex]y=a(x-(-1))(x-3)[/tex]
[tex]y=a(x+1)(x-3)[/tex] ...(i)
Put x=0 and y=3 because the y-intercept is (0,3).
[tex]3=a(0+1)(0-3)[/tex]
[tex]3=-3a[/tex]
[tex]\dfrac{3}{-3}=a[/tex]
[tex]-1=a[/tex]
Putting a=-1 in (i), we get
[tex]y=-1(x+1)(x-3)[/tex]
[tex]y=-(x+1)(x-3)[/tex]
Therefore, the equation of the parabola is [tex]y=-(x+1)(x-3)[/tex].
(b)
The vertex form of a parabola is
[tex]y=a(x-h)^2+k[/tex]
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.
[tex]y=a(x-2)^2+0[/tex]
[tex]y=a(x-2)^2[/tex] ...(ii)
Put x=0 and y=8 because the y-intercept is (0,8).
[tex]8=a(0-2)^2[/tex]
[tex]8=4a[/tex]
[tex]\dfrac{8}{4}=a[/tex]
[tex]2=a[/tex]
Putting a=2 in (ii), we get
[tex]y=2(x-2)^2[/tex]
Therefore, the equation of the parabola is [tex]y=2(x-2)^2[/tex].
