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Sagot :
Let's work through each of the given quadratic functions step-by-step to find the vertex, axis of symmetry, and [tex]\( y \)[/tex]-intercept.
### 3. [tex]\( f(x) = -3x^2 + 18x - 27 \)[/tex]
1. Vertex: The x-coordinate of the vertex of a quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] is given by the formula [tex]\[x = -\frac{b}{2a}\][/tex]
Here, [tex]\( a = -3 \)[/tex], [tex]\( b = 18 \)[/tex], and [tex]\( c = -27 \)[/tex].
[tex]\[ x = -\frac{18}{2(-3)} = 3 \][/tex]
Now, substitute [tex]\( x = 3 \)[/tex] back into the function to find the y-coordinate.
[tex]\[ f(3) = -3(3)^2 + 18(3) - 27 = -27 + 54 - 27 = 0 \][/tex]
So the vertex is at [tex]\( (3, 0) \)[/tex].
2. Axis of Symmetry: It is the vertical line that passes through the vertex.
[tex]\[ x = 3 \][/tex]
3. Y-intercept: The y-intercept is the value of the function when [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = -27 \][/tex]
So the y-intercept is [tex]\( -27 \)[/tex].
So for [tex]\( f(x) = -3x^2 + 18x - 27 \)[/tex]:
- Vertex: [tex]\( (3, 0) \)[/tex]
- Axis of Symmetry: [tex]\( x = 3 \)[/tex]
- [tex]\( y \)[/tex]-Intercept: [tex]\( -27 \)[/tex]
### 4. [tex]\( f(x) = x^2 - 8x + 19 \)[/tex]
1. Vertex:
[tex]\[ x = -\frac{-8}{2(1)} = 4 \][/tex]
Substitute [tex]\( x = 4 \)[/tex] back into the function:
[tex]\[ f(4) = (4)^2 - 8(4) + 19 = 16 - 32 + 19 = 3 \][/tex]
So the vertex is [tex]\( (4, 3) \)[/tex].
2. Axis of Symmetry:
[tex]\[ x = 4 \][/tex]
3. Y-intercept:
[tex]\[ f(0) = 19 \][/tex]
So the y-intercept is [tex]\( 19 \)[/tex].
So for [tex]\( f(x) = x^2 - 8x + 19 \)[/tex]:
- Vertex: [tex]\( (4, 3) \)[/tex]
- Axis of Symmetry: [tex]\( x = 4 \)[/tex]
- [tex]\( y \)[/tex]-Intercept: [tex]\( 19 \)[/tex]
### 5. [tex]\( f(x) = -2x^2 - 4x + 6 \)[/tex]
1. Vertex:
[tex]\[ x = -\frac{-4}{2(-2)} = 1 \][/tex]
Substitute [tex]\( x = 1 \)[/tex] back into the function:
[tex]\[ f(1) = -2(1)^2 - 4(1) + 6 = -2 - 4 + 6 = 0 \][/tex]
So the vertex is [tex]\( (1, 0) \)[/tex].
2. Axis of Symmetry:
[tex]\[ x = 1 \][/tex]
3. Y-intercept:
[tex]\[ f(0) = 6 \][/tex]
So the y-intercept is [tex]\( 6 \)[/tex].
So for [tex]\( f(x) = -2x^2 - 4x + 6 \)[/tex]:
- Vertex: [tex]\( (1, 0) \)[/tex]
- Axis of Symmetry: [tex]\( x = 1 \)[/tex]
- [tex]\( y \)[/tex]-Intercept: [tex]\( 6 \)[/tex]
### 6. [tex]\( P = -0.02n^2 + 3.40n - 16 \)[/tex] (Profit function)
1. Vertex:
[tex]\[ n = -\frac{3.40}{2(-0.02)} = -\frac{3.40}{-0.04} = 85 \][/tex]
Substitute [tex]\( n = 85 \)[/tex] back into the function:
[tex]\[ P(85) = -0.02(85)^2 + 3.40(85) - 16 = -0.02(7225) + 289 - 16 = -144.5 + 289 - 16 = 128.5 \][/tex]
The vertex, representing the maximum profit, is [tex]\( (85, 128.5) \)[/tex].
2. Axis of Symmetry:
[tex]\[ n = 85 \][/tex]
3. Y-intercept:
[tex]\[ P(0) = -16 \][/tex]
So the y-intercept is [tex]\( -16 \)[/tex].
So for [tex]\( P = -0.02n^2 + 3.40n - 16 \)[/tex]:
- Vertex: [tex]\( (85, 128.5) \)[/tex] (in thousands of units and thousands of dollars)
- Axis of Symmetry: [tex]\( n = 85 \)[/tex]
- [tex]\( y \)[/tex]-Intercept: [tex]\( -16 \)[/tex] (in thousands of dollars)
These steps summarize how to find the vertex, axis of symmetry, and [tex]\( y \)[/tex]-intercepts for each given quadratic function. Don't forget to sketch the graphs of the functions based on these characteristics!
### 3. [tex]\( f(x) = -3x^2 + 18x - 27 \)[/tex]
1. Vertex: The x-coordinate of the vertex of a quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] is given by the formula [tex]\[x = -\frac{b}{2a}\][/tex]
Here, [tex]\( a = -3 \)[/tex], [tex]\( b = 18 \)[/tex], and [tex]\( c = -27 \)[/tex].
[tex]\[ x = -\frac{18}{2(-3)} = 3 \][/tex]
Now, substitute [tex]\( x = 3 \)[/tex] back into the function to find the y-coordinate.
[tex]\[ f(3) = -3(3)^2 + 18(3) - 27 = -27 + 54 - 27 = 0 \][/tex]
So the vertex is at [tex]\( (3, 0) \)[/tex].
2. Axis of Symmetry: It is the vertical line that passes through the vertex.
[tex]\[ x = 3 \][/tex]
3. Y-intercept: The y-intercept is the value of the function when [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = -27 \][/tex]
So the y-intercept is [tex]\( -27 \)[/tex].
So for [tex]\( f(x) = -3x^2 + 18x - 27 \)[/tex]:
- Vertex: [tex]\( (3, 0) \)[/tex]
- Axis of Symmetry: [tex]\( x = 3 \)[/tex]
- [tex]\( y \)[/tex]-Intercept: [tex]\( -27 \)[/tex]
### 4. [tex]\( f(x) = x^2 - 8x + 19 \)[/tex]
1. Vertex:
[tex]\[ x = -\frac{-8}{2(1)} = 4 \][/tex]
Substitute [tex]\( x = 4 \)[/tex] back into the function:
[tex]\[ f(4) = (4)^2 - 8(4) + 19 = 16 - 32 + 19 = 3 \][/tex]
So the vertex is [tex]\( (4, 3) \)[/tex].
2. Axis of Symmetry:
[tex]\[ x = 4 \][/tex]
3. Y-intercept:
[tex]\[ f(0) = 19 \][/tex]
So the y-intercept is [tex]\( 19 \)[/tex].
So for [tex]\( f(x) = x^2 - 8x + 19 \)[/tex]:
- Vertex: [tex]\( (4, 3) \)[/tex]
- Axis of Symmetry: [tex]\( x = 4 \)[/tex]
- [tex]\( y \)[/tex]-Intercept: [tex]\( 19 \)[/tex]
### 5. [tex]\( f(x) = -2x^2 - 4x + 6 \)[/tex]
1. Vertex:
[tex]\[ x = -\frac{-4}{2(-2)} = 1 \][/tex]
Substitute [tex]\( x = 1 \)[/tex] back into the function:
[tex]\[ f(1) = -2(1)^2 - 4(1) + 6 = -2 - 4 + 6 = 0 \][/tex]
So the vertex is [tex]\( (1, 0) \)[/tex].
2. Axis of Symmetry:
[tex]\[ x = 1 \][/tex]
3. Y-intercept:
[tex]\[ f(0) = 6 \][/tex]
So the y-intercept is [tex]\( 6 \)[/tex].
So for [tex]\( f(x) = -2x^2 - 4x + 6 \)[/tex]:
- Vertex: [tex]\( (1, 0) \)[/tex]
- Axis of Symmetry: [tex]\( x = 1 \)[/tex]
- [tex]\( y \)[/tex]-Intercept: [tex]\( 6 \)[/tex]
### 6. [tex]\( P = -0.02n^2 + 3.40n - 16 \)[/tex] (Profit function)
1. Vertex:
[tex]\[ n = -\frac{3.40}{2(-0.02)} = -\frac{3.40}{-0.04} = 85 \][/tex]
Substitute [tex]\( n = 85 \)[/tex] back into the function:
[tex]\[ P(85) = -0.02(85)^2 + 3.40(85) - 16 = -0.02(7225) + 289 - 16 = -144.5 + 289 - 16 = 128.5 \][/tex]
The vertex, representing the maximum profit, is [tex]\( (85, 128.5) \)[/tex].
2. Axis of Symmetry:
[tex]\[ n = 85 \][/tex]
3. Y-intercept:
[tex]\[ P(0) = -16 \][/tex]
So the y-intercept is [tex]\( -16 \)[/tex].
So for [tex]\( P = -0.02n^2 + 3.40n - 16 \)[/tex]:
- Vertex: [tex]\( (85, 128.5) \)[/tex] (in thousands of units and thousands of dollars)
- Axis of Symmetry: [tex]\( n = 85 \)[/tex]
- [tex]\( y \)[/tex]-Intercept: [tex]\( -16 \)[/tex] (in thousands of dollars)
These steps summarize how to find the vertex, axis of symmetry, and [tex]\( y \)[/tex]-intercepts for each given quadratic function. Don't forget to sketch the graphs of the functions based on these characteristics!
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