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HOW DO YOU EXPLAIN THE ELEMENTS OF QUADRATIC FUNCTION BASED ON A GIVEN GRAPH AS YOUR ONLY SOURCE OF INFORMATION?

EXAMPLE I

Based on the graph given, find the value of h, p and q? Explain the plot of function by way of calculation

STEP 1 – GRAPH READING

1.1  FINDING THE MAXIMUM OR MINIMUM POINT

Since the graph comes in the shape of U, then it must have a minimum point. In this case, the minimum point is (-1, -8) where the axis symmetry is x =- 1

1.2  FINDING THE COORDINATE THAT INTERCEPTS WITH Y-AXIS

Based on the graph given, the y intercepts (f(x) intercept) lies on the point of -6 and has the coordinate of (0, -6)

1.3  FORMING THE FUNCTION/EQUATION BASED ON THE GIVEN COORDINATES

Since the minimum point is (-1, -8), therefore the function/equation shall read as follows :  f(x) = a(x + 1)²-8

Finding the value of a can be obtained upon taking the coordinate of y intercepts into the just formed function/equation :

At y intercepts, x = 0 and f(x) = – 6 and by replacing the value of f(x) and x into the function/equation will give new reading as follows :

– 6 = a(0 + 1)² – 8

– 6 = a – 8

  a  = – 6 + 8 = 2

  a = 2

As such, the complete reading of the function/equation shall read as :

 f(x) = 2(x + 1)² – 8

 

STEP 2 – FINDING THE VALUE OF h,p & q

Compare the complete function/equation with the given function/equation :

f(x) = h(x + p)² + q = 2(x + 1)² – 8

h = 2; p = 1; q = – 8

 

STEP 3 – FINDING THE ROOTS OR THE COORDINATES THAT INTERCEPTS WITH X-AXIS

The point where the function/equation intercepts with x-axis is/are also the roots of the function/equation. This shall be justified as finding the value of x when f(x) or y = 0 share the same mechanism as finding the roots of the function or equation.

Based on this understanding, the value of x shall be calculated as follows :

f(x) =   2(x + 1)² – 8 = 0

            2(x + 1)² = 8

            (x + 1)² = 8/2 = 4

            (x + 1) = + 2 or – 2

             x = 1 or – 3

Therefore, the x-axis intercept shall have 2 coordinates of (1, 0) and (-3, 0)

Finding the value of x can be done by other known methods such as applying factorization or completing the square or using the formula