Question

Write a quadratic function that opens down and goes through the point (5,0)

Answer 1

To write a quadratic function that opens down and goes through the point (5, 0), we can start with the general form of a quadratic function:

\[ f(x) = ax^2 + bx + c \]

Since we want the parabola to open downwards, we need \( a < 0 \).

Given the point (5, 0), we know that when \( x = 5 \), \( f(5) = 0 \). This means that we can use \( (5, 0) \) as a root of the quadratic. A quadratic function can be expressed in factored form using its roots:

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

where \( r_1 = 5 \) is one root. We can choose another root, say \( r_2 \). For simplicity, let's choose \( r_2 = 0 \), which gives us the quadratic function:

\[ f(x) = a(x - 5)(x - 0) = a(x - 5)x \]

To make sure the parabola opens downwards, we need \( a < 0 \). Let's choose \( a = -1 \) for simplicity:

\[ f(x) = -1(x - 5)x = -x^2 + 5x \]

So, one possible quadratic function that opens down and passes through the point (5, 0) is:

\[ f(x) = -x^2 + 5x \]

You can verify that this function opens down and includes the point (5, 0):

1. It opens down because \( a = -1 < 0 \).
2. When \( x = 5 \):

\[
f(5) = -5^2 + 5(5) = -25 + 25 = 0
\]

Thus, the quadratic function is valid.