Answer:
To do a full function study of y = x^4 - 8x^2 - 9, we need to determine the domain, intercepts, symmetry, asymptotes, intervals of increase and decrease, local extrema, and concavity.
Domain:
Since the function is a polynomial, it is defined for all real numbers. Therefore, the domain is (-∞, ∞).
x-Intercepts:
To find the x-intercepts, we set y = 0 and solve for x:
x^4 - 8x^2 - 9 = 0
We can factor the left-hand side to get:
(x^2 - 9)(x^2 + 1) = 0
This gives us x = ±√9 = ±3 as the x-intercepts.
y-Intercept:
To find the y-intercept, we set x = 0:
y = 0^4 - 8(0^2) - 9 = -9
Therefore, the y-intercept is (0, -9).
Symmetry:
The function is an even-degree polynomial, which means it has rotational symmetry of order 2 about the origin.
Asymptotes:
There are no vertical or horizontal asymptotes for this function.
Intervals of Increase and Decrease:
To find the intervals of increase and decrease, we need to find the critical points of the function by taking the first derivative and setting it equal to zero:
y' = 4x^3 - 16x = 0
Solving for x, we get x = 0 or x = ±√4 = ±2. Therefore, the critical points are (-2, 43), (0, -9), and (2, 43). We can use the second derivative test to determine that (-2, 43) and (2, 43) are local minima and (0, -9) is a local maximum.
The function increases on the intervals (-∞, -2) and (2, ∞) and decreases on the interval (-2, 2).
Local Extrema:
The local minimum points are (-2, 43) and (2, 43), and the local maximum point is (0, -9).
Concavity:
To determine the concavity of the function, we take the second derivative:
y'' = 12x^2 - 16
Setting y'' equal to zero, we get x = ±√4/3. Since y'' is positive for x < -√4/3 and x > √4/3, and negative for -√4/3 < x < √4/3, we have a point of inflection at x = -√4/3 and x = √4/3.
Plotting the Graph:
We can now use all of the information we have gathered to sketch the graph of y = x^4 - 8x^2 - 9. The graph has rotational symmetry of order 2 about the origin, and it passes through the points (-3, 0), (0, -9), and (3, 0). It has local minimum points at (-2, 43) and (2, 43) and a local maximum point at (0, -9). It changes concavity at x = -√4/3 and x = √4/3. Here is a rough sketch of the graph:
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