Question

Write the equation of the parabola that passes through the points (2,0), (
–
1,0), and (3,8).
Write your answer in the form y=a(x–p)(x–q), where a, p, and q are integers, decimals, or simplified fractions.

Answer 1

To find the equation of the parabola that passes through the given points (2,0), (-1,0), and (3,8), we can utilize the fact that the equation of a parabola can be expressed in the vertex form as \(y = a(x - p)(x - q)\), where \(p\) and \(q\) are the x-coordinates of the x-intercepts.

Given the points (2, 0) and (-1, 0), we can identify the roots (x-intercepts) of the parabola:
- \(p = 2\) (since \(y = 0\) when \(x = 2\))
- \(q = -1\) (since \(y = 0\) when \(x = -1\))

Now, we can write the equation in the form:

\[
y = a(x - 2)(x + 1)
\]

Next, we need to determine the value of \(a\). To do this, we can use the third point (3, 8) to find \(a\).

Substituting \(x = 3\) and \(y = 8\) into the equation:

\[
8 = a(3 - 2)(3 + 1)
\]

This simplifies to:

\[
8 = a(1)(4) \implies 8 = 4a
\]

To solve for \(a\):

\[
a = \frac{8}{4} = 2
\]

Now that we have \(a\), we can finalize the equation of the parabola:

\[
y = 2(x - 2)(x + 1)
\]

Thus, the equation of the parabola is:

\[
\boxed{y = 2(x - 2)(x + 1)}
\]