Question

What is the function of f(x) = x2 + 10x + 9

Answer 1

`\((-5, -16)\).`

Explanation:

Let's break down the function
`\(f(x) = x^2 + 10x + 9\)` in simpler terms.

1. Start with the function
`\(f(x) = x^2 + 10x + 9\).`

2. This is a quadratic function because it has a term with
`\(x^2\)`, and it's called quadratic because it represents a parabola.

3. You can factor this quadratic expression as:

`\[f(x) = (x + 9)(x + 1)\]`

This means that
`\(f(x)\)` is the result of multiplying two simpler expressions
`\((x + 9)\)` and
`\((x + 1)\)`together.

4. When you have a quadratic expression in this factored form, you can see where it equals zero. For this function, it equals zero when
`\(x = -9\) (from \((x + 9) = 0\)) and when \(x = -1\) (from \((x + 1) = 0\)).`

5. These values of
`\(x\)` where
`\(f(x) = 0\)` are called the "roots" or "zeroes" of the function. They are the points where the parabola crosses the x-axis.

6. The shape of the parabola depends on the coefficient of the
`\(x^2\)`term. In this case, since the coefficient is positive (1), the parabola opens upward.

7. The vertex of the parabola (the highest or lowest point) can be found using the formula
`\(x_{\text{vertex}} = (-b)/(2a)\),`where
`\(a\)` is the coefficient of the
`\(x^2\)`term (1) and
`\(b\)` is the coefficient of the
`\(x\)` term (10). Plugging in these values, you find
`\(x_{\text{vertex}} = -5\).`

8. To find the corresponding
`\(y\)`-coordinate of the vertex, plug
`\(x_{\text{vertex}} = -5\)`into the original function:

`\[f(-5) = (-5)^2 + 10(-5) + 9 = 25 - 50 + 9 = -16\]`

9. So, the vertex of the parabola is at
`\((-5, -16)\)`, and the function
`\(f(x)\)`describes this quadratic relationship. It represents a parabola that opens upward with its vertex at
`\((-5, -16)\).`