Which statements are true about the graph of the function f(x) = 6x^4 – x^2?

The function f(x) = 6x⁴ – x² is a polynomial function, and its graph has some important characteristics:

• Degree and Leading Coefficient: The degree of the polynomial is 4, and the leading coefficient is positive (6). This means that as x approaches positive or negative infinity, the graph will rise towards positive infinity on both ends.
• Y-intercept: To find the y-intercept, we can evaluate f(0). Substituting 0 in the function gives us f(0) = 6(0)⁴ – (0)² = 0. Thus, the y-intercept is at the origin (0, 0).
• Symmetry: Since the function contains only even powers of x (4 and 2), it is an even function. This means that the graph is symmetric about the y-axis.
• Critical Points: To determine the critical points where the graph changes direction, we can take the derivative of the function and set it to zero. Taking the derivative, f'(x) = 24x³ – 2x, and setting it equal to zero allows us to find the x-values where the slope is zero. Factoring gives us 2x(12x² – 1) = 0, leading to critical points that provide information about local maxima and minima.

In summary, the function f(x) = 6x⁴ – x² has a y-intercept at (0, 0), is symmetric about the y-axis, and its graph rises toward positive infinity as x approaches both positive and negative infinity. Understanding these characteristics can help in graphing the function accurately.