IDNLearn.com offers a unique blend of expert answers and community-driven insights. Our community provides accurate and timely answers to help you understand and solve any issue.
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.
We 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. Your search for answers ends at IDNLearn.com. Thank you for visiting, and we hope to assist you again soon.
