Sketch the graph f(x)=-2x^2(x-5)^3(x+2)^4.
What would be my coordinates. I solved it and my graph looks nothing like the one on wolframalpha. Can someone show me how you would solve this?
sketch?
What do you mean you solved it?
Well I need to graph it but it days I did it wrong. I don't know what to do with the 2 in front.
I made a sign chart and got
+,+,+,- and used the roots -2,0,5 and grape it but I think I'm making a mistake somewhere.
To sketch the graph of the function f(x) = -2x^2(x-5)^3(x+2)^4, we can follow these steps:
1. Determine the x-intercepts:
To find the x-intercepts of the graph, set f(x) equal to zero and solve for x:
-2x^2(x-5)^3(x+2)^4 = 0
The x-intercepts occur when any of the factors equals zero. So, we set each factor equal to zero and solve separately:
-2x^2 = 0 => x = 0 (multiplicity 2)
x-5 = 0 => x = 5 (multiplicity 1)
x+2 = 0 => x = -2 (multiplicity 1)
Therefore, the x-intercepts are (0,0), (5,0), and (-2,0).
2. Determine the behavior as x approaches negative or positive infinity:
Since the term -2x^2 has the highest degree, the behavior of the graph as x approaches infinity or negative infinity will be the same as -2x^2.
When x is very large positive or negative, -2x^2 will be very large negative. Therefore, the left and right edges of the graph will approach negative infinity.
3. Determine the end behavior at x = -2 and x = 5:
For x < -2 and x > 5, the negative exponent in (x+2)^4 and (x-5)^3 will cause these factors to approach infinity. So, the graph will approach positive infinity as x approaches -2 and 5.
4. Determine the behavior near each x-intercept:
For x close to 0, the term -2x^2 approaches 0, while the factors (x-5)^3 and (x+2)^4 remain positive. So, the graph will approach the x-axis, passing through the point (0,0).
For x close to 5, the term (x-5)^3 approaches 0 from the positive side, while the other factors remain positive. So, the graph will pass through (5,0) from below.
The behavior near x = -2 is similar to x = 5, but the graph passes through (-2,0) from above.
5. Determine the concavity of the graph:
The graph will change concavity at the zeros, -2 and 5. To determine the concavity, we can examine the factors (x-5)^3 and (x+2)^4. Both factors are powers of odd and even numbers, respectively, which means (x-5)^3 is negative for x < 5 and positive for x > 5, while (x+2)^4 is positive for all x. Therefore, the graph will be concave downwards to the left of x = 5 and concave upwards to the right of x = 5. Likewise, it will be concave upwards to the left of x = -2 and concave downwards to the right of x = -2.
By following these steps, you should be able to sketch the graph of f(x) = -2x^2(x-5)^3(x+2)^4 accurately.