Question

Write a quadratic function that opens up and goes through the points (-3,0) and (4,0)

Answer 1

To write a quadratic function that opens upwards and passes through the points \((-3, 0)\) and \( (4, 0) \), we can start by noting that these points are the roots of the quadratic function.

The general form of a quadratic function with roots \(r_1\) and \(r_2\) can be expressed as:

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

In this case, the roots are \(r_1 = -3\) and \(r_2 = 4\). Plugging these values into the formula gives:

\[
f(x) = a(x + 3)(x - 4)
\]

Now, we need to determine the value of \(a\). Since we want the quadratic to open upwards, we can choose \(a > 0\).

For simplicity, we can set \(a = 1\):

\[
f(x) = (x + 3)(x - 4)
\]

Expanding this:

\[
f(x) = x^2 - 4x + 3x - 12
\]
\[
f(x) = x^2 - x - 12
\]

Thus, one quadratic function that opens upward and passes through the points \((-3, 0)\) and \((4, 0)\) is:

\[
f(x) = x^2 - x - 12
\]

You could also multiply by any positive constant \(a\) to get different quadratic functions that have the same roots, for example:

\[
f(x) = 2(x + 3)(x - 4) = 2x^2 - 2x - 24
\]

But the simplest function is:

\[
f(x) = x^2 - x - 12
\]