The equation y = 2x^4 - 1 defines a relationship between x and y, where x is the input and y is the output. Which statements about the relationship are true? Select all that apply. Multi-select Choices: (A)The graph is curved. (B)The y-intercept is (-1,0). (C)When the input is 2, the output is 31. (D)The function is linear. (E)The rate of change is not constant.

Question

The equation y = 2x^4 - 1 defines a relationship between x and y, where x is the input and y is the output. Which statements about the relationship are true? Select all that apply.   Multi-select Choices: (A)The graph is curved. (B)The y-intercept is (-1,0). (C)When the input is 2, the output is 31. (D)The function is linear. (E)The rate of change is not constant.

Bytelearn · Accepted Answer

Determine Graph Shape:  Analyze the equation y = 2x^4 - 1 to determine the graph's shape. Since the highest power of x is 4, which is even, the graph is symmetric and curved like a parabola but steeper. Identify Y-Intercept:  Identify the y-intercept by substituting x = 0 into the equation. y = 2(0)^4 - 1 = -1. The y-intercept is (0, -1), not (-1, 0). Calculate Output for x=2:  Calculate the output when the input x is 2. y = 2(2)^4 - 1 = 2(16) - 1 = 32 - 1 = 31. This confirms that when x is 2, y is indeed 31. Check Linearity:  Determine if the function is linear by checking if the equation is of the form y = mx + b. Since the equation is y = 2x^4 - 1, it is not linear because it involves x to the fourth power. Evaluate Rate of Change:  Evaluate if the rate of change is constant by considering the derivative. The derivative of y = 2x^4 - 1 is dy/dx = 8x^3, which depends on the value of x, indicating that the rate of change is not constant.

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