Answered

IDNLearn.com: Where questions are met with accurate and insightful answers. Find reliable solutions to your questions quickly and accurately with help from our dedicated community of experts.

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 :

GinnyAnsweridn
To determine the vertex of the quadratic function [tex]\( f(x) = x^2 - 18x + 60 \)[/tex], we can follow these steps:

### Step 1: Finding the [tex]\( x \)[/tex]-coordinate of the vertex
The [tex]\( x \)[/tex]-coordinate of the vertex for a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]

For the given function:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = -18 \][/tex]
[tex]\[ c = 60 \][/tex]

Plugging in the values, we get:
[tex]\[ x = -\frac{-18}{2 \times 1} \][/tex]
[tex]\[ x = \frac{18}{2} \][/tex]
[tex]\[ x = 9 \][/tex]

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

So, the vertex of the function is at [tex]\( (9, -21) \)[/tex].

### Step 3: Analyzing the given statements
Let's evaluate each statement based on our calculations:

1. The [tex]\( x \)[/tex]-coordinate of the vertex is greater than the [tex]\( y \)[/tex]-coordinate.
- Here, [tex]\( x = 9 \)[/tex] and [tex]\( y = -21 \)[/tex].
- Since [tex]\( 9 \)[/tex] is greater than [tex]\( -21 \)[/tex], this statement is true.

2. The [tex]\( x \)[/tex]-coordinate of the vertex is negative.
- The [tex]\( x \)[/tex]-coordinate is [tex]\( 9 \)[/tex], which is not negative. Therefore, this statement is false.

3. The [tex]\( y \)[/tex]-coordinate of the vertex is greater than the [tex]\( y \)[/tex]-intercept.
- The [tex]\( y \)[/tex]-intercept is the [tex]\( y \)[/tex]-value when [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 0^2 - 18 \times 0 + 60 = 60 \][/tex]
- The [tex]\( y \)[/tex]-coordinate of the vertex is [tex]\( -21 \)[/tex], which is not greater than [tex]\( 60 \)[/tex]. Therefore, this statement is false.

4. The [tex]\( y \)[/tex]-coordinate of the vertex is positive.
- The [tex]\( y \)[/tex]-coordinate is [tex]\( -21 \)[/tex], which is not positive. Therefore, this statement is false.

### Conclusion
The statement about the vertex of the function that is true is:

- The [tex]\( x \)[/tex]-coordinate of the vertex is greater than the [tex]\( y \)[/tex]-coordinate.
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com is your reliable source for answers. We appreciate your visit and look forward to assisting you again soon.