Question

Determine the graph behavior at the zero(s) of the polynomial function f left parenthesis x right parenthesis equals x squared minus 6 x plus 9.
(1 point)
Responses

The graph of the function touches the x-axis at x equals negative 3 and x = 3.
The graph of the function touches the x -axis at Image with alt text: x equals negative 3 and x = 3.

The graph of the function crosses the x-axis at x equals negative 3 and x = 3.
The graph of the function crosses the x -axis at Image with alt text: x equals negative 3 and x = 3.

The graph of the function touches the x-axis at x = 3.
The graph of the function touches the x -axis at x = 3.

The graph of the function crosses the x-axis at x equals negative 3.

Answer 1

To analyze the behavior of the polynomial function \( f(x) = x^2 - 6x + 9 \), we first need to find its zeros.

1. **Factor the polynomial:**
\[
f(x) = (x - 3)^2
\]

2. **Find the zeros:**
Setting the polynomial equal to zero, we have:
\[
(x - 3)^2 = 0
\]
This gives:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
So there is a repeated root at \( x = 3 \).

3. **Behavior at the zero:**
Since the factor is squared, the graph will **touch** the x-axis at the zero and will not cross it.

Given this analysis, the correct response is:

**The graph of the function touches the x-axis at \( x = 3 \).**