Find the Quadratic equation of function (f) whose graph is shown below.

Find the Quadratic equation of function (f) whose graph is shown below.

Find the Quadratic equation of function (f) whose graph is shown above.

Step By Step solution :-

Let be quadratic equation is

y=f(x)=ax^{2}+bx+c: 

f(x) passing through points are (-1,4),(-3,12) and (0,6):

Calculate the value of c:

y=f(x)=ax^{2}+bx+c: at\ x=0\ and\ y=6

Solving it , we get:

y=f(0)=a\cdot(0)^{2}+b\cdot(0)+c=6\\ \implies y=6=c

So ,

c=6

Simplify the expression

y=ax^{2}+bx+c:\ at\ y=4,x=-1,c=6 

Putting value of x=-1, y=4 and c=6 and in f(x), we get:

y=f(-1)=a\cdot(-1)^{2}+b\cdot(-1)+6=4\\ \implies a-b+6=4\\ \implies a=b-2

Similarly, Simplify the Expression for

y=ax^{2}+bx+c:\ at\ x=-3,y=12,c=6

Putting value of x=-3, y=12 and c=6 and in f(x), we get:

y=f(-3)=a\cdot(-3)^{2}+b\cdot(-3)+6=12\\ \implies 9 \cdot a- 3\cdot b+6=12\\  \implies 3\cdot a- b+2=4\\ \implies 3\cdot a= b+2

Simplify the expression

a=b-2\\ 3⋅a=b+2

Substitute value of a in subsequent equation

3⋅(b-2)=b+2\implies 3⋅b-6=b+2 \implies2\cdot b =8\\ \ \\ \therefore b=4

Calculate the value of a for b=4:

a=b-2\\ \implies a=4-2\implies a=2

Hence, equation of quadratic equation is:

y=f(x)=2x^{2}+4x+6

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