Question

write the equation of the parabola that passes through the points shown in the tab
x | y
-10 | 14
-8 | 0
-3 | 0
write your answer in the form y = a(x - p)(x - q), where a, p, and q are integers, de
or simplified fractions.

Explanation:

Step1: Identify roots from table

From the table, when \( y = 0 \), \( x=-8 \) and \( x = - 3 \). So the roots (zeros) of the parabola are \( x=-8 \) and \( x=-3 \). In the factored form \( y=a(x - p)(x - q) \), if the roots are \( r_1\) and \( r_2\), then \( p=-r_1\) and \( q=-r_2\)? Wait, no. Wait, the factored form is \( y=a(x - p)(x - q) \), so if the roots are \( x = p \) and \( x = q \) when \( y = 0 \). Wait, no: if \( x=-8 \) is a root, then \( x+8=0 \), so \( (x + 8) \) is a factor. Similarly, \( x=-3 \) is a root, so \( (x + 3) \) is a factor. So we can write \( y=a(x + 8)(x + 3) \). Now we need to find the value of \( a \). We can use the other point \( (-10,14) \). Substitute \( x=-10 \) and \( y = 14 \) into the equation:

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

Substitute \( x=-10 \), \( y = 14 \) into \( y=a(x + 8)(x + 3) \):

\( 14=a(-10 + 8)(-10 + 3) \)

Simplify the terms inside the parentheses:

\( -10 + 8=-2 \) and \( -10 + 3=-7 \)

So the equation becomes:

\( 14=a(-2)(-7) \)

Calculate \( (-2)(-7)=14 \)

So \( 14=a\times14 \)

Divide both sides by 14:

\( a = 1 \)

Step3: Write the final equation

Now that we have \( a = 1 \), \( p=-8 \) (wait, no, in our factored form we had \( y=a(x + 8)(x + 3) \), which is \( y=1\times(x - (-8))(x - (-3)) \). So in the form \( y=a(x - p)(x - q) \), \( p=-8 \) and \( q=-3 \), and \( a = 1 \). So the equation is \( y=(x + 8)(x + 3) \) or \( y = 1(x - (-8))(x - (-3)) \)

Answer:

\( y=(x + 8)(x + 3) \) (or \( y = 1(x - (-8))(x - (-3)) \))