Answer :
Certainly! Let's analyze the function [tex]\( y = x^4 - 31x^2 - 180 \)[/tex] step by step to find the intercepts and test for symmetry.
### Intercepts
#### Y-Intercept
The y-intercept happens where the function crosses the y-axis. To find the y-intercept, we set [tex]\( x = 0 \)[/tex] in the function.
[tex]\[ y = (0)^4 - 31(0)^2 - 180 \][/tex]
[tex]\[ y = -180 \][/tex]
So, the y-intercept is [tex]\((0, -180)\)[/tex].
#### X-Intercepts
The x-intercepts occur where the function crosses the x-axis. To find the x-intercepts, we set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex].
[tex]\[ 0 = x^4 - 31x^2 - 180 \][/tex]
This equation needs to be solved for [tex]\( x \)[/tex]. The solutions to this equation are:
[tex]\[ x = -6, \quad x = 6, \quad x = -\sqrt{5}i, \quad x = \sqrt{5}i \][/tex]
However, [tex]\( -\sqrt{5}i \)[/tex] and [tex]\( \sqrt{5}i \)[/tex] are complex numbers and thus, do not represent real intercepts on the Cartesian plane. The real x-intercepts are:
[tex]\[ x = -6 \quad \text{and} \quad x = 6 \][/tex]
So, the real x-intercepts are [tex]\((-6, 0)\)[/tex] and [tex]\((6, 0)\)[/tex].
### Symmetry
Let's test for symmetry of the function around the y-axis and the origin:
#### Symmetry around the Y-Axis
A function is symmetric around the y-axis if [tex]\( f(x) = f(-x) \)[/tex].
[tex]\[ y = x^4 - 31x^2 - 180 \][/tex]
[tex]\[ y(-x) = (-x)^4 - 31(-x)^2 - 180 \][/tex]
[tex]\[ y(-x) = x^4 - 31x^2 - 180 \][/tex]
Since [tex]\( y(x) = y(-x) \)[/tex], the function is symmetric around the y-axis.
#### Symmetry around the X-Axis
Symmetry around the x-axis would require [tex]\( f(x) = -f(x) \)[/tex], but this is not generally the case for functions defined as [tex]\( y = f(x) \)[/tex].
#### Symmetry around the Origin
A function is symmetric around the origin if [tex]\( f(x) = -f(-x) \)[/tex].
[tex]\[ y = x^4 - 31x^2 - 180 \][/tex]
[tex]\[ -y(-x) = -(x^4 - 31x^2 - 180) \][/tex]
[tex]\[ -y(-x) = -x^4 + 31x^2 + 180 \][/tex]
Since [tex]\( y(x) \neq -y(-x) \)[/tex], the function is not symmetric around the origin.
### Conclusion
- Y-Intercept: [tex]\((0, -180)\)[/tex]
- X-Intercepts: [tex]\((-6, 0)\)[/tex] and [tex]\((6, 0)\)[/tex]
- Symmetry:
- The function is symmetric around the y-axis.
- The function is not symmetric around the origin.
### Intercepts
#### Y-Intercept
The y-intercept happens where the function crosses the y-axis. To find the y-intercept, we set [tex]\( x = 0 \)[/tex] in the function.
[tex]\[ y = (0)^4 - 31(0)^2 - 180 \][/tex]
[tex]\[ y = -180 \][/tex]
So, the y-intercept is [tex]\((0, -180)\)[/tex].
#### X-Intercepts
The x-intercepts occur where the function crosses the x-axis. To find the x-intercepts, we set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex].
[tex]\[ 0 = x^4 - 31x^2 - 180 \][/tex]
This equation needs to be solved for [tex]\( x \)[/tex]. The solutions to this equation are:
[tex]\[ x = -6, \quad x = 6, \quad x = -\sqrt{5}i, \quad x = \sqrt{5}i \][/tex]
However, [tex]\( -\sqrt{5}i \)[/tex] and [tex]\( \sqrt{5}i \)[/tex] are complex numbers and thus, do not represent real intercepts on the Cartesian plane. The real x-intercepts are:
[tex]\[ x = -6 \quad \text{and} \quad x = 6 \][/tex]
So, the real x-intercepts are [tex]\((-6, 0)\)[/tex] and [tex]\((6, 0)\)[/tex].
### Symmetry
Let's test for symmetry of the function around the y-axis and the origin:
#### Symmetry around the Y-Axis
A function is symmetric around the y-axis if [tex]\( f(x) = f(-x) \)[/tex].
[tex]\[ y = x^4 - 31x^2 - 180 \][/tex]
[tex]\[ y(-x) = (-x)^4 - 31(-x)^2 - 180 \][/tex]
[tex]\[ y(-x) = x^4 - 31x^2 - 180 \][/tex]
Since [tex]\( y(x) = y(-x) \)[/tex], the function is symmetric around the y-axis.
#### Symmetry around the X-Axis
Symmetry around the x-axis would require [tex]\( f(x) = -f(x) \)[/tex], but this is not generally the case for functions defined as [tex]\( y = f(x) \)[/tex].
#### Symmetry around the Origin
A function is symmetric around the origin if [tex]\( f(x) = -f(-x) \)[/tex].
[tex]\[ y = x^4 - 31x^2 - 180 \][/tex]
[tex]\[ -y(-x) = -(x^4 - 31x^2 - 180) \][/tex]
[tex]\[ -y(-x) = -x^4 + 31x^2 + 180 \][/tex]
Since [tex]\( y(x) \neq -y(-x) \)[/tex], the function is not symmetric around the origin.
### Conclusion
- Y-Intercept: [tex]\((0, -180)\)[/tex]
- X-Intercepts: [tex]\((-6, 0)\)[/tex] and [tex]\((6, 0)\)[/tex]
- Symmetry:
- The function is symmetric around the y-axis.
- The function is not symmetric around the origin.
