Certainly! Let's break down and write a detailed step-by-step explanation for the given expression [tex]\( y = 2x^3 - 3x + 2 \)[/tex].
1. Understanding the Function:
- The function [tex]\( y = 2x^3 - 3x + 2 \)[/tex] is a polynomial of degree 3.
- This means it is a cubic function, as the highest power of [tex]\( x \)[/tex] is 3.
2. Identifying the Components:
- The term [tex]\( 2x^3 \)[/tex] is the cubic term.
- The term [tex]\( -3x \)[/tex] is the linear term.
- The constant [tex]\( 2 \)[/tex] is a constant term not dependent on [tex]\( x \)[/tex].
3. Graphing the Function:
- To graph this polynomial function, we need to identify key characteristics such as the intercepts, critical points, and inflection points.
4. Finding the Intercepts:
- Y-intercept: This is found by setting [tex]\( x = 0 \)[/tex].
[tex]\[
y = 2(0)^3 - 3(0) + 2 = 2
\][/tex]
Thus, the y-intercept is [tex]\( (0, 2) \)[/tex].
- X-intercepts: These are found by solving the equation [tex]\( 2x^3 - 3x + 2 = 0 \)[/tex]. Finding the roots of this cubic equation can be complex and often requires numerical methods or factoring techniques. For simplicity, we can note the general process, although finding the exact roots is beyond the scope of this immediate explanation.
5. Finding Critical Points (Local Extrema):
- To find the critical points, take the first derivative of [tex]\( y \)[/tex]:
[tex]\[
y' = \frac{d}{dx} (2x^3 - 3x + 2) = 6x^2 - 3
\][/tex]
- Set the derivative equal to zero to find the critical points:
[tex]\[
6x^2 - 3 = 0
\][/tex]
[tex]\[
6x^2 = 3
\][/tex]
[tex]\[
x^2 = \frac{1}{2}
\][/tex]
[tex]\[
x = \pm \sqrt{\frac{1}{2}} = \pm \frac{\sqrt{2}}{2}
\][/tex]
- These values of [tex]\( x \)[/tex] give us the critical points at:
[tex]\[
y = 2 \left( \frac{\sqrt{2}}{2} \right)^3 - 3 \left( \frac{\sqrt{2}}{2} \right) + 2 \quad \text{and} \quad y = 2 \left( -\frac{\sqrt{2}}{2} \right)^3 - 3 \left( -\frac{\sqrt{2}}{2} \right) + 2
\][/tex]
6. Finding Inflection Points:
- To find inflection points, take the second derivative:
[tex]\[
y'' = \frac{d}{dx} (6x^2 - 3) = 12x
\][/tex]
- Set the second derivative equal to zero to find inflection points:
[tex]\[
12x = 0
\][/tex]
[tex]\[
x = 0
\][/tex]
- Substitute [tex]\( x = 0 \)[/tex] back into the original function to find the corresponding [tex]\( y \)[/tex]-value. We find:
[tex]\[
y = 2(0)^3 - 3(0) + 2 = 2
\][/tex]
Thus, the inflection point is at [tex]\( (0, 2) \)[/tex].
By following these steps, we have examined key features of the cubic polynomial [tex]\( y = 2x^3 - 3x + 2 \)[/tex], including intercepts, critical points, and the inflection point. This helps us understand the shape and behavior of the function more comprehensively.
