Question

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

x	y
-2	0
-1	-8
0	-14
3	-20
4	-18
7	0

Explanation:

Step1: Identify roots from table

From the table, when \( y = 0 \), \( x=-2 \) and \( x = 7 \). So the roots of the quadratic function are \( x=-2 \) and \( x = 7 \). A quadratic function with roots \( r_1\) and \( r_2\) can be written in factored form as \( y=a(x - r_1)(x - r_2) \). Substituting the roots, we get \( y=a(x + 2)(x - 7) \).

Step2: Find the value of \( a \)

We can use another point from the table to find \( a \). Let's use the point \( (0,-14) \) (when \( x = 0 \), \( y=-14 \)). Substitute \( x = 0 \) and \( y=-14 \) into the equation \( y=a(x + 2)(x - 7) \):

$$ -14=a(0 + 2)(0 - 7) $$

Simplify the right - hand side: \( a\times2\times(-7)=-14a \)
So, \( -14=-14a \)
Divide both sides by \( -14 \): \( a = 1 \)

Step3: Write the final equation

Substitute \( a = 1 \) into the factored form \( y=a(x + 2)(x - 7) \). Expand the equation:

$$
LATEXBLOCK0
$$

Answer:

The equation of the quadratic function is \( y=x^2-5x - 14 \)