Question

1. write the equation of the quadratic function which passes through the points in the table shown below.

x	y
1	0
2	-6
4	-12
5	-12
6	-10
8	0

Explanation:

Step1: Identify roots from table

From the table, when \( x = 1 \) and \( x = 8 \), \( y = 0 \). So the roots are \( r_1 = 1 \) and \( r_2 = 8 \). The quadratic function can be written in factored form as \( y = a(x - 1)(x - 8) \).

Step2: Find the value of \( a \)

Use the point \( (0, 8) \) (when \( x = 0 \), \( y = 8 \)) to substitute into the factored form:
\( 8 = a(0 - 1)(0 - 8) \)
\( 8 = a(-1)(-8) \)
\( 8 = 8a \)
Divide both sides by 8: \( a = 1 \).

Step3: Expand the factored form

Substitute \( a = 1 \) into \( y = (x - 1)(x - 8) \) and expand:
\( y = x^2 - 8x - x + 8 \)
\( y = x^2 - 9x + 8 \). We can verify with other points (e.g., \( x = 2 \): \( y = 4 - 18 + 8 = -6 \), which matches the table).

Answer:

\( y = x^2 - 9x + 8 \)