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Sagot :
Certainly! Let's solve and analyze the function [tex]\( y = x^2 - 4x - 1 \)[/tex] step-by-step.
### Step 1: Understand the Function
The given function is a quadratic equation:
[tex]\[ y = x^2 - 4x - 1 \][/tex]
### Step 2: Identify the Coefficients
In the quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex], we can identify the coefficients:
- [tex]\( a = 1 \)[/tex] (coefficient of [tex]\( x^2 \)[/tex])
- [tex]\( b = -4 \)[/tex] (coefficient of [tex]\( x \)[/tex])
- [tex]\( c = -1 \)[/tex] (constant term)
### Step 3: Determine the Vertex of the Parabola
A quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] is a parabola. The vertex (h, k) of the parabola can be found using:
[tex]\[ h = -\frac{b}{2a} \][/tex]
[tex]\[ k = f(h) \][/tex]
Let's calculate [tex]\( h \)[/tex]:
[tex]\[ h = -\frac{-4}{2 \cdot 1} = \frac{4}{2} = 2 \][/tex]
To find [tex]\( k \)[/tex], substitute [tex]\( x = 2 \)[/tex] into the equation:
[tex]\[ k = (2)^2 - 4(2) - 1 \][/tex]
[tex]\[ k = 4 - 8 - 1 = -5 \][/tex]
So, the vertex of the parabola is [tex]\( (2, -5) \)[/tex].
### Step 4: Find the Axis of Symmetry
The axis of symmetry for the parabola is the vertical line that passes through the vertex. Thus, the axis of symmetry is:
[tex]\[ x = 2 \][/tex]
### Step 5: Determine the Direction of the Parabola
Since the coefficient [tex]\( a = 1 \)[/tex] is positive, the parabola opens upwards.
### Step 6: Find the x-intercepts
To find the x-intercepts, we set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = x^2 - 4x - 1 \][/tex]
To solve this quadratic equation, we can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = -1 \)[/tex]. Plugging in these values:
[tex]\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{16 + 4}}{2} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{20}}{2} \][/tex]
[tex]\[ x = \frac{4 \pm 2\sqrt{5}}{2} \][/tex]
[tex]\[ x = 2 \pm \sqrt{5} \][/tex]
So, the x-intercepts are:
[tex]\[ x = 2 + \sqrt{5} \][/tex]
[tex]\[ x = 2 - \sqrt{5} \][/tex]
### Step 7: Find the y-intercept
The y-intercept is found by setting [tex]\( x = 0 \)[/tex] in the equation:
[tex]\[ y = (0)^2 - 4(0) - 1 = -1 \][/tex]
So, the y-intercept is at [tex]\( (0, -1) \)[/tex].
### Summary of the Function Analysis:
1. Vertex: [tex]\( (2, -5) \)[/tex]
2. Axis of Symmetry: [tex]\( x = 2 \)[/tex]
3. Direction: Opens upwards
4. x-intercepts: [tex]\( x = 2 + \sqrt{5} \)[/tex] and [tex]\( x = 2 - \sqrt{5} \)[/tex]
5. y-intercept: [tex]\( (0, -1) \)[/tex]
Thus, the function [tex]\( y = x^2 - 4x - 1 \)[/tex] describes a parabola with the above characteristics.
### Step 1: Understand the Function
The given function is a quadratic equation:
[tex]\[ y = x^2 - 4x - 1 \][/tex]
### Step 2: Identify the Coefficients
In the quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex], we can identify the coefficients:
- [tex]\( a = 1 \)[/tex] (coefficient of [tex]\( x^2 \)[/tex])
- [tex]\( b = -4 \)[/tex] (coefficient of [tex]\( x \)[/tex])
- [tex]\( c = -1 \)[/tex] (constant term)
### Step 3: Determine the Vertex of the Parabola
A quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] is a parabola. The vertex (h, k) of the parabola can be found using:
[tex]\[ h = -\frac{b}{2a} \][/tex]
[tex]\[ k = f(h) \][/tex]
Let's calculate [tex]\( h \)[/tex]:
[tex]\[ h = -\frac{-4}{2 \cdot 1} = \frac{4}{2} = 2 \][/tex]
To find [tex]\( k \)[/tex], substitute [tex]\( x = 2 \)[/tex] into the equation:
[tex]\[ k = (2)^2 - 4(2) - 1 \][/tex]
[tex]\[ k = 4 - 8 - 1 = -5 \][/tex]
So, the vertex of the parabola is [tex]\( (2, -5) \)[/tex].
### Step 4: Find the Axis of Symmetry
The axis of symmetry for the parabola is the vertical line that passes through the vertex. Thus, the axis of symmetry is:
[tex]\[ x = 2 \][/tex]
### Step 5: Determine the Direction of the Parabola
Since the coefficient [tex]\( a = 1 \)[/tex] is positive, the parabola opens upwards.
### Step 6: Find the x-intercepts
To find the x-intercepts, we set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = x^2 - 4x - 1 \][/tex]
To solve this quadratic equation, we can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = -1 \)[/tex]. Plugging in these values:
[tex]\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{16 + 4}}{2} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{20}}{2} \][/tex]
[tex]\[ x = \frac{4 \pm 2\sqrt{5}}{2} \][/tex]
[tex]\[ x = 2 \pm \sqrt{5} \][/tex]
So, the x-intercepts are:
[tex]\[ x = 2 + \sqrt{5} \][/tex]
[tex]\[ x = 2 - \sqrt{5} \][/tex]
### Step 7: Find the y-intercept
The y-intercept is found by setting [tex]\( x = 0 \)[/tex] in the equation:
[tex]\[ y = (0)^2 - 4(0) - 1 = -1 \][/tex]
So, the y-intercept is at [tex]\( (0, -1) \)[/tex].
### Summary of the Function Analysis:
1. Vertex: [tex]\( (2, -5) \)[/tex]
2. Axis of Symmetry: [tex]\( x = 2 \)[/tex]
3. Direction: Opens upwards
4. x-intercepts: [tex]\( x = 2 + \sqrt{5} \)[/tex] and [tex]\( x = 2 - \sqrt{5} \)[/tex]
5. y-intercept: [tex]\( (0, -1) \)[/tex]
Thus, the function [tex]\( y = x^2 - 4x - 1 \)[/tex] describes a parabola with the above characteristics.
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