Answered

Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Our Q&A platform provides quick and trustworthy answers to your questions from experienced professionals in different areas of expertise. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

Charla wants to determine the vertex of the function [tex][tex]$f(x) = x^2 - 18x + 60$[/tex][/tex] by changing the function into vertex form. Which statement about the vertex of the function is true?

A. The [tex][tex]$x$[/tex][/tex]-coordinate of the vertex is greater than the [tex][tex]$y$[/tex][/tex]-coordinate.
B. The [tex][tex]$x$[/tex][/tex]-coordinate of the vertex is negative.
C. The [tex][tex]$y$[/tex][/tex]-coordinate of the vertex is greater than the [tex][tex]$y$[/tex][/tex]-intercept.
D. The [tex][tex]$y$[/tex][/tex]-coordinate of the vertex is positive.

Sagot :

GinnyAnswer
Certainly! Let's determine the vertex of the function [tex]\( f(x) = x^2 - 18x + 60 \)[/tex] step-by-step and analyze the statements provided.

### Step 1: Find the Vertex of the Parabola
The function given is a quadratic equation in the form [tex]\( f(x) = ax^2 + bx + c \)[/tex] where [tex]\( a = 1 \)[/tex], [tex]\( b = -18 \)[/tex], and [tex]\( c = 60 \)[/tex].

The vertex form of a quadratic equation can be derived using the formula for the x-coordinate of the vertex:
[tex]\[ x = \frac{-b}{2a} \][/tex]

Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x = \frac{-(-18)}{2 \cdot 1} = \frac{18}{2} = 9 \][/tex]

So, the x-coordinate of the vertex is [tex]\( x = 9 \)[/tex].

### Step 2: Calculate the y-coordinate of the Vertex
To find the y-coordinate of the vertex, substitute [tex]\( x = 9 \)[/tex] back into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(9) = (9)^2 - 18 \cdot 9 + 60 \][/tex]
[tex]\[ f(9) = 81 - 162 + 60 \][/tex]
[tex]\[ f(9) = -21 \][/tex]

So, the y-coordinate of the vertex is [tex]\( y = -21 \)[/tex].

### Step 3: Find the y-Intercept of the Function
The y-intercept of the function occurs when [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = (0)^2 - 18 \cdot 0 + 60 = 60 \][/tex]

So, the y-intercept is [tex]\( y = 60 \)[/tex].

### Step 4: Analyze the Statements
1. The x-coordinate of the vertex is greater than the y-coordinate.
[tex]\[ 9 > -21 \][/tex]
This statement is True.

2. The x-coordinate of the vertex is negative.
[tex]\[ 9 \][/tex] (The x-coordinate) is not negative.
This statement is False.

3. The y-coordinate of the vertex is greater than the y-intercept.
[tex]\[ -21 > 60 \][/tex]
This is False.

4. The y-coordinate of the vertex is positive.
[tex]\[ -21 \][/tex]
This is not positive.
This statement is False.

### Conclusion
Based on our detailed analysis, the statement that is true about the vertex of the function is:
The x-coordinate of the vertex is greater than the y-coordinate.

Therefore, the correct statement is:
The x-coordinate of the vertex is greater than the y-coordinate.
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.