Question

write the equation of the parabola that passes through the points shown in the table.
x | y
-5 | -24
-3 | 0
-2 | 0
write your answer in the form y = a(x - p)(x - q), where a, p, and q are integers, decimals, or simplified fractions.

Explanation:

Step1: Identify roots from table

From the table, when \( y = 0 \), \( x=-3 \) and \( x = -2 \). So the roots are \( p=-3 \), \( q=-2 \). The factored form is \( y=a(x + 3)(x + 2) \) (since \( x - p=x-(-3)=x + 3 \), \( x - q=x-(-2)=x + 2 \)).

Step2: Find 'a' using a point

Use the point \( (-5,-24) \). Substitute \( x=-5 \), \( y=-24 \) into \( y=a(x + 3)(x + 2) \):

$$ -24=a(-5 + 3)(-5 + 2) $$

Simplify the right - hand side: \( (-5 + 3)=-2 \), \( (-5 + 2)=-3 \), so \( -24=a(-2)(-3)=6a \).

Step3: Solve for 'a'

Divide both sides by 6: \( a=\frac{-24}{6}=-4 \).

Step4: Write the equation

Substitute \( a = - 4 \), \( p=-3 \), \( q=-2 \) into \( y=a(x - p)(x - q) \), we get \( y=-4(x + 3)(x + 2) \).

Answer:

\( y=-4(x + 3)(x + 2) \)