# 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 :-

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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