Answer :
1)
Given,
The coordinates of the vertex, (h, k)=(2, -4).
The coordinates of a point through which the graph passes, (x, y)=(0,0).
The vertex form of the quadratic equation for the graph is,
[tex]y=a(x-h)^2+k\ldots\ldots.(1)[/tex]Here, (h, k) is the coordinates of the vertex of the graph.
Put the values of h, k, x and y in the above equation to find the value of a.
[tex]\begin{gathered} 0=a(0-2)^2-4 \\ 0=a\times4-4 \\ 4=4a \\ \frac{4}{4}=a \\ 1=a \end{gathered}[/tex]Now, put the values of h, k and a in equation (1) to obtain the vertex form of the quadratic equation whose graph has vertex (2, -4) and passing through point (0, 0).
[tex]\begin{gathered} y=1\times(x-2)^2-4 \\ y=(x-2)^2-4 \end{gathered}[/tex]So, the vertex form of the quadratic equation whose graph has vertex (2, -4) and passing through point (0, 0) is,
[tex]y=(x-2)^2-4[/tex]2)
From the graph, the coordinates of a point through which the graph passes is (x,y)=(-2, -4).
The intercept form of a quadratic equation is,
[tex]y=a(x-p)(x-q)\text{ ------(1)}[/tex]Here, p and q are the x intercepts.
From the graph, the x intercepts are p=-3 and q=2.
Now, put the values of x, y, p and q in the above equation to find the value of a.
[tex]\begin{gathered} -4=a(-2-(-3))(-2-2) \\ -4=a(-2+3)(-4) \\ -4=a\times1\times(-4) \\ a=1 \end{gathered}[/tex]Now, put the a values of. a, p and q in equation (2) to find the intercept form of quadratic function of the graph.
[tex]\begin{gathered} y=1\times(x-(-3))(x-2) \\ y=(x+3)(x-2) \end{gathered}[/tex]Therefore, the intercept form of quadratic function of the graph is y=(x+3)(x-2).
