Alright, let's break down the function step by step.
We have the function \( y = 3x^4 - 2x^2 + 8 \). This is a polynomial function consisting of three terms: \( 3x^4 \), \( -2x^2 \), and \( 8 \).
### Step-by-Step Analysis
1. Identify the components:
- The first term is \( 3x^4 \). This term suggests a quartic (fourth degree) polynomial.
- The second term is \( -2x^2 \). This term suggests a quadratic (second degree) polynomial.
- The third term is \( 8 \). This is a constant term.
2. Analyze the individual terms:
- \( 3x^4 \):
- Coefficient: 3
- Degree: 4
- \( -2x^2 \):
- Coefficient: -2
- Degree: 2
- \( 8 \):
- This is a constant term and has no \( x \) associated with it.
3. Construct the polynomial:
The polynomial combines all these terms into a single expression:
[tex]\[
y = 3x^4 - 2x^2 + 8
\][/tex]
4. Put it all together:
When we combine the terms, our function doesn't change:
[tex]\[
y = 3x^4 - 2x^2 + 8
\][/tex]
Given the function \( y = 3x^4 - 2x^2 + 8 \), we can also discuss some of the properties of this function:
- Degree of the polynomial: The highest exponent of \( x \) in the function is 4, so the degree of this polynomial is 4.
- Leading coefficient: The coefficient of the term with the highest degree (in this case \( x^4 \)) is 3.
Graphically, this function would show a typical quartic curve, potentially with local minima and maxima. However, since the dominant term \( 3x^4 \) is positive, the ends of the graph will go upwards as \( x \) approaches \( \pm \infty \).
In summary, the unpacked and detailed function is:
[tex]\[
y = 3x^4 - 2x^2 + 8
\][/tex]
