Question

please help me solve this question

f(x)= Use the graph of the quadratic function f to write its formula as f(x) = a(x - 1)2 + k. Q Q 4- LY 2- x -6 -4 -2 4 6 -2

step by step

Answer 1

We know that the expression for polynomial function f(x) having roots at a,b,c..... is given by f(x)=k(x-a)(x-b)(x-c)..... where k is the leading coefficient of the polynomial function. Also, a quadratic function is a polynomial function and if a polynomial function has zeros having even multiplicity then at that point the graph does NOT cross x axis and if it has odd multiplicity, then it passes x axis.

From the given graph, we observe the following things:

f(x) has 0, 2 as the zeros.

At both 0 and 2, the graph crosses x axis and hence it has zeros having odd multiplicities. Since we have to find the quadratic function, the zeros will be 0 and 2.


The graph of quadratic function passes through the point (1,5), hence the polynomial must satisfy the given point.
Now by using the above observations, we may write the expression for polynomial function f(x) as:-

f(x)=k(x-0)(x-2)

=> f(x)=k(x)(x-2)

f(x) must satisfy (1,5)
=> 5=k(1)(1-2)
=> k=-5

Hence the polynomial function is:-

=> f(x)=-5(x)(x-2)

Now, we have to represent the above function in the form a(x-h)²+k

=> f(x)=-5(x)(x-2)

=> f(x)=-5(x²-2x)

=> f(x)=-5{(x)²-2(x)(1)+1²-1²}

=> f(x)=-5{(x-1)²-1}

=> f(x)=-5(x-1)²+5

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