can someone show how to graph step by step

the polynomial function -x^4 - 4x^2. That is -x to the fourth power minus -4x squared. Please Help!!!

Method 1: use a graphing tool and enter the function.

Method 2: Using several values for x, evaluate the function and plot the points (x,f(x)) to see the shape of the curve.

If you can't evaluate f(x), then you might reconsider taking pre-calculus.

As a starting point, note that f = -x^2(x^2+4), so you know there is a double root at x=0, so the graph is tangent to the x-axis at (0,0). Since it's a quartic, as x gets large, f gets large negative. Since there are no other real roots, the graph may look kind of like a flattened parabola opening downward. That will be the case here if f(x) < 0 for x near 0. Indeed, f(x) < 0 near x=0, so we know the graph is tangent to the x-axis from below.

FYI: Quartics may also look like a rounded W shape, But since the graph opens down, and is tangent below the x-axis, there can't be any bumps nearby, because then the graph would cross the x-axis near x=0. That would mean there are real roots.

Sure! I can help you with that. I will walk you through the steps to graph the polynomial function -x^4 - 4x^2.

Step 1: Determine the x-intercepts
To find the x-intercepts (where the graph crosses or touches the x-axis), set the function equal to zero and solve for x. In this case, set -x^4 - 4x^2 = 0 and factor the equation if possible.

-x^4 - 4x^2 = 0
x^2(-x^2 - 4) = 0

Now we have two possibilities:
1) x^2 = 0, which gives x = 0.
2) -x^2 - 4 = 0. Rearranging the equation, we get -x^2 = 4. Solving for x^2, we have x^2 = -4, which has no real solutions.

Therefore, the x-intercept for this function is x = 0.

Step 2: Determine the y-intercept
To find the y-intercept (where the graph crosses or touches the y-axis), substitute x = 0 into the function and solve.

f(x) = -x^4 - 4x^2
f(0) = -(0)^4 - 4(0)^2
f(0) = 0

So, the y-intercept is y = 0.

Step 3: Determine the symmetry of the graph
To check for symmetry, we need to determine if the function is even or odd. If the function is even, then it is symmetric with respect to the y-axis. If it is odd, then it is symmetric with respect to the origin.

In this case, the function -x^4 - 4x^2 is an even function because it only contains even powers of x. Therefore, it is symmetric with respect to the y-axis.

Step 4: Determine the end behavior
The end behavior describes the values of y as x approaches infinity or negative infinity. For this polynomial function, the end behavior can be determined by looking at the power of the highest degree term, which is -x^4.

We know that when the power is even (positive), the curve of the graph opens upwards on both ends, and when the power is odd (negative), the curve opens downwards.

In this case, since the power is even and negative (-x^4), the graph will open upwards, resembling an "U" shape.

Step 5: Plot additional points
To sketch the graph accurately, we can plot a few more points to get a better understanding of its shape. Choose some x-values within a reasonable range (say, -3, -2, -1, 1, 2, and 3) and substitute them into the function to find the corresponding y-values.

For example:
f(-3) = -(-3)^4 - 4(-3)^2 = -81 - 36 = -117
f(-2) = -(-2)^4 - 4(-2)^2 = -16 - 16 = -32
f(-1) = -(-1)^4 - 4(-1)^2 = -1 - 4 = -5
f(1) = -(1)^4 - 4(1)^2 = -1 - 4 = -5
f(2) = -(2)^4 - 4(2)^2 = -16 - 16 = -32
f(3) = -(3)^4 - 4(3)^2 = -81 - 36 = -117

Now, you can plot these points on a graph, connecting them smoothly to complete the shape of the curve. Make sure to include the x-intercept and the y-intercept we found earlier.

The graph of the polynomial function -x^4 - 4x^2 would resemble a downward-opening "U" shape symmetric with respect to the y-axis, and it will pass through the points (-3, -117), (-2, -32), (-1, -5), (0, 0), (1, -5), (2, -32), and (3, -117).

I hope this explanation helps you graph the given polynomial function accurately. Let me know if you have any further questions!