charla wants to determine the vertex of the function $f(x) = x^2 - 18x + 60$ by changing the function into…

Explanation:

Step1: Complete the square for $x$

Start with $f(x) = x^2 - 18x + 60$. Take half of $-18$: $\frac{-18}{2} = -9$, square it: $(-9)^2 = 81$. Rewrite the function: $$f(x) = (x^2 - 18x + 81) - 81 + 60$$

Step2: Simplify to vertex form

Factor the perfect square trinomial and combine constants: $$f(x) = (x - 9)^2 - 21$$ Vertex form is $f(x) = a(x-h)^2 + k$, where vertex is $(h,k)$.

Step3: Identify vertex and check statements

Vertex is $(9, -21)$.

• $x$-coordinate $9$ is positive, so first statement is false.
• $y$-coordinate $-21$ is negative, so second statement is false.
• $y$-intercept is $f(0)=60$, $-21 < 60$, so third statement is false. (Note: The fourth option is cut off, but based on the visible options, none of the shown ones are true. However, if we assume the question expects evaluating the provided options, the only conclusion is that the visible options are false, but if we focus on the vertex calculation)

Answer:

None of the visible statements are true. The vertex of the function is $(9, -21)$.

• The $x$-coordinate of the vertex is positive (not negative).
• The $y$-coordinate of the vertex is negative (not positive).
• The $y$-coordinate of the vertex $(-21)$ is less than the $y$-intercept $(60)$.