use the methods of this section to sketch the curve. y=x^3-3a^2x+2a^3, where a is a positive constant. What do the members of this family of curves have in common? How do they differ from each other?
Factor
x^3-3*a^2*x+2*a^3
to get
(x-a)²(x+2a)
What does the factor (x-a)² indicate?
What abuot (x+2a)?
To sketch the curve of the equation \(y = x^3 - 3a^2x + 2a^3\), where \(a\) is a positive constant, we can follow these steps:
Step 1: Find the y-intercept.
To find the y-intercept, substitute \(x = 0\) into the equation:
\(y = (0)^3 - 3a^2(0) + 2a^3\)
\(y = 2a^3\)
So, the y-intercept is at the point (0, \(2a^3\)).
Step 2: Find the x-intercept(s).
To find the x-intercepts, set \(y\) equal to zero and solve for \(x\):
\(0 = x^3 - 3a^2x + 2a^3\)
This equation can be factored as follows:
\(0 = (x - a)(x^2 + ax - 2a^2)\)
To find the x-intercepts, we solve for \(x\) in each factor:
\(x - a = 0\), so \(x = a\) (x-intercept at (a, 0)).
\(x^2 + ax - 2a^2 = 0\) is a quadratic equation, and we can solve it using factoring, completing the square, or the quadratic formula. It factors as \((x - a)(x + 2a) = 0\), so x = a and x = -2a are the x-intercepts.
Step 3: Find the turning point(s).
To find the turning point(s), we need to find the x-value(s) where the derivative of \(y\) with respect to \(x\) is equal to zero.
Differentiating \(y\) with respect to \(x\), we get:
\(\frac{dy}{dx} = 3x^2 - 3a^2\)
Setting \(\frac{dy}{dx} = 0\), we have:
\(3x^2 - 3a^2 = 0\)
Dividing both sides by 3, we get:
\(x^2 - a^2 = 0\)
This equation factors as:
\((x - a)(x + a) = 0\)
So, x = a and x = -a are the x-values of the turning points.
Step 4: Plot the points obtained and sketch the curve.
- Plot the y-intercept at (0, \(2a^3\)).
- Plot the x-intercepts at (a, 0) and (-2a, 0).
- Plot the turning points at (a, \(a^3 - 3a^2\)) and (-a, \(-a^3 - 3a^2\)).
Now, with the points plotted, you can sketch the curve by connecting the points smoothly. The shape of the curve will depend on the value of \(a\).
Common features of this family of curves:
- The y-intercept is always at (0, \(2a^3\)).
- The x-intercepts are at (a, 0) and (-2a, 0).
- The turning points are at (a, \(a^3 - 3a^2\)) and (-a, \(-a^3 - 3a^2\)).
Differences between the curves in this family:
- The values of the parameters \(a\) determine the specifics of each curve, such as the steepness, the position of the turning points, and the distances between the intercepts.
To sketch the curve of the equation y = x^3 - 3a^2x + 2a^3, where "a" is a positive constant, we can follow these steps:
Step 1: Find the intercepts:
To find the x-intercepts, set y = 0 and solve for x:
x^3 - 3a^2x + 2a^3 = 0
Step 2: Factorize or solve for x:
This equation might not factorize easily, so you might need to use numerical methods or graphing calculators to solve for x.
Step 3: Find the y-intercept:
To find the y-intercept, set x = 0 and solve for y:
y = 0^3 - 3a^2(0) + 2a^3
y = 2a^3
Step 4: Plot the points:
- Plot the x-intercepts and the y-intercept on a coordinate system.
Step 5: Determine the behavior of the curve:
- Observe the degree of the polynomial function, which is 3. This means that the curve will have a specific behavior known as an "odd function".
- If "a" is positive, the curve will have different behavior from when "a" is negative.
What members of this family of curves have in common:
1. All curves in this family have a y-intercept of 2a^3.
How they differ from each other:
1. The x-intercepts of the curves depend on the particular value of "a".
2. The behavior of the curves will differ depending on whether "a" is positive or negative. Specifically, when "a" is positive, the curve will have a downward-pointing shape, and when "a" is negative, the curve will have an upward-pointing shape.
3. As "a" varies, the scale and position of the curves will change. For larger values of "a", the curves will be stretched vertically, and for smaller values, they will be compressed.
4. The overall shape of the curves will remain similar, except for the changes mentioned above.