Question

Write the equation of the parabola that passes through the points (
–
1,0), (2,0), and (1,5).
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 points \((-1, 0)\), \((2, 0)\), and \((1, 5)\), we can start by noting that the points \((-1, 0)\) and \((2, 0)\) are the x-intercepts of the parabola. Therefore, we can express the parabola in factored form as:

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

Next, to find the value of \(a\), we will use the third point \((1, 5)\). We substitute \(x = 1\) and \(y = 5\) into the equation:

\[
5 = a(1 + 1)(1 - 2)
\]

Calculating the expressions in the parentheses:

\[
5 = a(2)(-1)
\]

This simplifies to:

\[
5 = -2a
\]

To solve for \(a\):

\[
a = -\frac{5}{2}
\]

Now, we can write down the final equation for the parabola:

\[
y = -\frac{5}{2}(x + 1)(x - 2)
\]

Thus, the equation of the parabola in the requested form \(y = a(x - p)(x - q)\) is:

\[
y = -\frac{5}{2}(x + 1)(x - 2)
\]

Here, \(a = -\frac{5}{2}\), \(p = -1\), and \(q = 2\).