Visualize for f(x) = x⁴ - 8x² + 7, identifying all x-intercepts and local extrema.

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Let's analyze the function f of x equals x to the fourth minus 8 x squared plus 7. This quartic function has a W-shape with symmetry about the y-axis. We can identify four x-intercepts at negative square root of 7, negative 1, positive 1, and positive square root of 7. The function has a local maximum at the origin with a value of 7, and two local minima at x equals negative 2 and positive 2, both with a value of negative 9. To find the x-intercepts, we set the function equal to zero and solve for x. We have x to the fourth minus 8 x squared plus 7 equals zero. This is a quadratic in terms of x squared, so we can substitute u equals x squared. This gives us u squared minus 8u plus 7 equals zero, which factors as (u minus 1) times (u minus 7) equals zero. Solving for u, we get u equals 1 or u equals 7. Substituting back, we have x squared equals 1 or x squared equals 7, which gives us x equals plus or minus 1, or x equals plus or minus square root of 7. These four values are our x-intercepts. To find the critical points, we take the derivative of the function and set it equal to zero. The derivative is 4x cubed minus 16x, which can be factored as 4x times x squared minus 4, or 4x times (x minus 2) times (x plus 2). Setting this equal to zero, we get x equals 0, or x equals plus or minus 2. To determine whether these critical points are maxima or minima, we use the second derivative test. The second derivative is 12x squared minus 16. At x equals negative 2, the second derivative is 32, which is positive, indicating a local minimum. At x equals 0, the second derivative is negative 16, indicating a local maximum. At x equals 2, the second derivative is again 32, indicating another local minimum. So we have a local maximum at (0, 7) and local minima at (-2, -9) and (2, -9). Let's examine the behavior and symmetry of our function. First, notice that f of negative x equals f of x for all x, which means this is an even function. We can verify this algebraically: f of negative x equals negative x to the fourth minus 8 times negative x squared plus 7, which simplifies to x to the fourth minus 8x squared plus 7, which is exactly f of x. This even function property means the graph is symmetric about the y-axis, as we can see. The domain of this function is all real numbers, and its range is from negative 9 to positive infinity. Since the leading term is x to the fourth with a positive coefficient, the function approaches positive infinity as x approaches either positive or negative infinity. The overall shape resembles the letter W, with a central peak at (0, 7) and two valleys at (-2, -9) and (2, -9). Let's summarize what we've learned about the function f of x equals x to the fourth minus 8x squared plus 7. This quartic polynomial has a distinctive W-shape. We found four x-intercepts at x equals plus or minus 1 and x equals plus or minus square root of 7 by factoring the equation as (x squared minus 1) times (x squared minus 7) equals zero. The function has three critical points: a local maximum at x equals 0 with value 7, and two local minima at x equals plus or minus 2, both with value negative 9. The function is even, meaning f of negative x equals f of x, which creates the symmetry about the y-axis that we observed. The domain of this function is all real numbers, and its range is from negative 9 to positive infinity. This analysis demonstrates how calculus techniques can reveal the complete behavior of polynomial functions.