Question

Write a quadratic function that's opens down using the points (1,0) and (7,0)

Answer 1

To write a quadratic function that opens downward and has roots at the points (1, 0) and (7, 0), we can start with the factored form of a quadratic function:

\[
f(x) = a(x - r_1)(x - r_2)
\]

where \( r_1 \) and \( r_2 \) are the roots of the function. Since the roots are (1, 0) and (7, 0), we have:

\[
f(x) = a(x - 1)(x - 7)
\]

To make the quadratic open downward, we need to choose \( a < 0 \). For simplicity, let's choose \( a = -1 \):

\[
f(x) = -1(x - 1)(x - 7)
\]

Now let's expand this expression:

\[
f(x) = -1[(x - 1)(x - 7)]
\]
\[
= -1[x^2 - 7x - x + 7]
\]
\[
= -1[x^2 - 8x + 7]
\]
\[
= -x^2 + 8x - 7
\]

Thus, the quadratic function that opens downward and has roots at (1, 0) and (7, 0) is:

\[
f(x) = -x^2 + 8x - 7
\]