Question

write the equation of the parabola that passes through the points shown in the table.

x	y
-2	-14
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

The parabola has roots at \( x = -4 \) and \( x = 2 \) (since \( y = 0 \) at these \( x \)-values). So, the factored form is \( y = a(x - (-4))(x - 2) = a(x + 4)(x - 2) \).

Step2: Substitute a point to find \( a \)

Use the point \( (-2, -14) \). Substitute \( x = -2 \) and \( y = -14 \) into the equation:
\( -14 = a(-2 + 4)(-2 - 2) \)
Simplify: \( -14 = a(2)(-4) \)
\( -14 = -8a \)
Solve for \( a \): \( a = \frac{-14}{-8} = \frac{7}{4} \)? Wait, no—wait, calculation error. Wait, \( (-2 + 4) = 2 \), \( (-2 - 2) = -4 \), so \( 2 \times -4 = -8 \). Then \( -14 = a(-8) \), so \( a = \frac{-14}{-8} = \frac{7}{4} \)? Wait, no, wait the table: \( x=-2, y=-14 \). Wait, maybe I made a mistake. Wait, let's recheck. Wait, the roots are at \( x = -4 \) (so \( x + 4 = 0 \)) and \( x = 2 \) (so \( x - 2 = 0 \)). So the equation is \( y = a(x + 4)(x - 2) \). Now plug in \( x = -2 \), \( y = -14 \):

\( -14 = a(-2 + 4)(-2 - 2) \)
\( -14 = a(2)(-4) \)
\( -14 = -8a \)
Divide both sides by -8: \( a = \frac{14}{8} = \frac{7}{4} \)? Wait, but that's not an integer. Wait, maybe I misread the table. Wait the table: \( x=-4, y=0 \); \( x=-2, y=-14 \); \( x=2, y=0 \). Wait, maybe the roots are correct, but let's check again. Wait, maybe the form is \( y = a(x - p)(x - q) \), so if roots are at \( x = -4 \) and \( x = 2 \), then \( p = -4 \), \( q = 2 \), so \( y = a(x - (-4))(x - 2) = a(x + 4)(x - 2) \). Then plug in \( x = -2 \), \( y = -14 \):

\( -14 = a(-2 + 4)(-2 - 2) \)
\( -14 = a(2)(-4) \)
\( -14 = -8a \)
So \( a = \frac{14}{8} = \frac{7}{4} \). But the problem says \( a, p, q \) are integers, decimals, or simplified fractions. Wait, maybe I made a mistake in the roots. Wait, maybe the roots are at \( x = -4 \) and \( x = 2 \), so \( p = -4 \), \( q = 2 \), and \( a = \frac{7}{4} \)? But that's a fraction. Wait, no, wait the table: \( x=-2, y=-14 \). Wait, maybe I miscalculated. Wait, \( (-2 + 4) = 2 \), \( (-2 - 2) = -4 \), so \( 2 \times -4 = -8 \). Then \( -14 = a \times -8 \), so \( a = 14/8 = 7/4 \). But the problem says "integers, decimals, or simplified fractions". So \( a = 7/4 \)? Wait, but that seems odd. Wait, maybe the roots are different? Wait, no, the points \( (-4, 0) \) and \( (2, 0) \) are zeros, so those are the roots. So the equation is \( y = \frac{7}{4}(x + 4)(x - 2) \)? Wait, but let's check with \( x = -2 \): \( \frac{7}{4}(2)(-4) = \frac{7}{4}(-8) = -14 \), which matches. So that's correct. Wait, but the problem says "integers, decimals, or simplified fractions", so \( a = 7/4 \), \( p = -4 \), \( q = 2 \). So the equation is \( y = \frac{7}{4}(x + 4)(x - 2) \). Wait, but maybe I made a mistake in the sign. Wait, the form is \( y = a(x - p)(x - q) \), so if the root is at \( x = -4 \), then \( x - p = x - (-4) = x + 4 \), so \( p = -4 \); root at \( x = 2 \), so \( x - q = x - 2 \), so \( q = 2 \). So that's correct. Then \( a = 7/4 \). So the equation is \( y = \frac{7}{4}(x + 4)(x - 2) \).

Wait, but let me recheck the calculation. When \( x = -2 \), \( (x + 4) = 2 \), \( (x - 2) = -4 \), so \( 2 \times -4 = -8 \). Then \( a \times -8 = -14 \), so \( a = (-14)/(-8) = 14/8 = 7/4 \). Yes, that's correct. So the equation is \( y = \frac{7}{4}(x + 4)(x - 2) \).

Answer:

\( y = \frac{7}{4}(x + 4)(x - 2) \)