Question

For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.

Answer 1

Consider the table provided in the problem.

Substitute the points (0, 1), (1, 4) and (-1, 0) in the standard form of quadratic equation.

     f(x) = a(x – h)2 + k

        1 = a(0 – h)2 + k

       1 = ah2 + k

-k + 1 = ah2 ...... (1)

 

Now substitute (1, 4) in the standard form.

4 = a(1 – h)2 + k

4 = a – 2ah + ah2 + k

 

From (1)

4 = a – 2ah – k + 1 + k

3 = a – 2ah ...... (2)

 

Now substitute (-1, 0) in the standard form.

0 = a(-1 – h)2 + k

0 = a + 2ah + ah2 + k

 

From (1)

 0 = a + 2ah – k + 1 + k

-1 = a + 2ah ...... (3)

 

Add (2) and (3)

3 – 1 = a – 2ah + a + 2ah

      2 = 2a

      a = 1

 

Substitute value of ‘a’ in (2)

3 = 1 – 2(1)h

2 = -2h

h = -1

 

Put in (1)

-k + 1 = 1(-1)2

       k = 0

 

Substitute the values of a, h and k in the standard form.

f(x) = 1{x – (-1)}2 + 0

f(x) = x2 + 2x + 1

Hence, the required general equation is f(x) = x2 + 2x + 1.

Hence, the required general equation is f(x) = x2 + 2x + 1.