Question

Постройте график функций y=x^4-8x^2+16

Answer 1

Ответ смотри во вложении

Answer 2

Graph of the Function y=x^4-8x^2+16

To graph the function y=x^4-8x^2+16, we can start by analyzing its key features and then plotting the graph.

Key Features of the Function

1. Degree of the Polynomial: The given function is a polynomial of degree 4. 2. Turning Points: The function may have turning points where the gradient is zero. 3. Intercepts: We can find the x-intercepts by setting y=0 and solving for x. 4. End Behavior: As x approaches positive or negative infinity, the behavior of the function can be determined.

Graphing the Function

Let's start by finding the turning points and intercepts of the function.

Turning Points

The turning points of the function can be found by taking the derivative of the function and solving for x where the derivative is equal to 0.

The derivative of the function y=x^4-8x^2+16 is: ``` y' = 4x^3 - 16x ``` Setting y' to 0 and solving for x: ``` 4x^3 - 16x = 0 4x(x^2 - 4) = 0 x(x+2)(x-2) = 0 ``` So, the turning points occur at x=0, x=-2, and x=2.

Intercepts

To find the x-intercepts, we set y=0 and solve for x: ``` x^4-8x^2+16 = 0 ``` This is a quadratic equation in terms of x^2. We can solve for x^2 and then find the corresponding values of x.

End Behavior

As x approaches positive or negative infinity, the behavior of the function can be determined by the leading term of the polynomial.

After analyzing these key features, we can plot the graph of the function y=x^4-8x^2+16.

I hope this helps! Let me know if you need further assistance.