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Sagot :
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.
### 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.
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