Determine the maximum possible number of turning points for the graph of the ntion.
g(x)=-1/5x + 2
I got 0
How do I graph f(x)=x^5-4x^3-12x
your first one is right
for f(x)=x^5-4x^3-12x
=x(x^4 - 4x^2 - 12)
=x(x^2-6)(x^2+2)
if this had been the equation
x^5-4x^3-12x = 0
there would be solutions of
x=0, x=±√6 = ±2.45, and 2 complex roots
so the graph crosses at -2.45, 0, 2,45
it rises into the first quadrant, and drops into the third quadrants.
The two complex roots cause a "wiggle" in the first quadrant above the x-axis
To determine the maximum possible number of turning points for a function, you can use the degree of the polynomial. The degree tells you the highest power of x in the function.
For the function g(x) = -1/5x + 2, you are correct in saying that it has 0 turning points. This is because it is a linear function with a degree of 1, and linear functions do not have any turning points. They either have a positive or negative slope.
Now, let's talk about graphing the function f(x) = x^5 - 4x^3 - 12x.
To graph this function, you can follow these steps:
1. Determine the x-intercepts: Set the function equal to zero and solve for x. These are the points where the graph crosses the x-axis. In this case, set f(x) = 0:
x^5 - 4x^3 - 12x = 0
Factoring out an "x," we get:
x(x^4 - 4x^2 - 12) = 0
Setting each factor equal to zero, we get:
x = 0 (this is one x-intercept)
x^4 - 4x^2 - 12 = 0
The equation x^4 - 4x^2 - 12 = 0 is a quadratic equation in terms of x^2. You can solve it by factoring, completing the square, or using the quadratic formula.
2. Determine the y-intercept: Substitute x = 0 into the function f(x) to find the corresponding y-value. In this case, we have:
f(0) = (0)^5 - 4(0)^3 - 12(0) = 0
Therefore, the y-intercept is (0,0).
3. Plot additional points: Choose some x-values within a specific range and substitute them into the function to find the corresponding y-values. This will help you plot more points and see the overall shape of the graph. You can choose enough points to get a clear picture of the graph.
4. Determine the end behavior: As x approaches positive or negative infinity, you can analyze the leading term of the function to determine how the graph behaves. In this case, the leading term is x^5. Since the degree is odd (5 is an odd number), the end behavior will be similar to that of a cubic function.
Once you have all this information, plot the points on a coordinate plane and connect them smoothly to obtain the graph of the function f(x) = x^5 - 4x^3 - 12x.
Note: If you have access to graphing software or online graphing calculators, you can also use them to graph the function and get a more accurate representation.