sketch the graph of f(x)= -x^2 (x+1) (x-2)
thanks!
you know there are zeroes at x = 0,-1,2
This is a cubic with a negative leading coefficient, so it goes up to the left and down to the right.
So start off, then make it wiggle on the x-axis so it goes through the roots.
Done.
To sketch the graph of the function f(x) = -x^2(x+1)(x-2), we can follow these steps:
Step 1: Find the x-intercepts
To find the x-intercepts, we set f(x) equal to zero and solve for x.
- Set -x^2(x+1)(x-2) = 0
- We can see that there are three factors: -x^2 = 0, (x+1) = 0, and (x-2) = 0.
• For -x^2 = 0, the only solution is x = 0.
• For (x+1) = 0, the solution is x = -1.
• For (x-2) = 0, the solution is x = 2.
So, we have three x-intercepts: x = 0, x = -1, and x = 2.
Step 2: Determine the y-intercept
To find the y-intercept, we substitute x = 0 into the equation and solve for f(x).
- Substitute x = 0 into the equation: f(0) = -(0^2)(0+1)(0-2)
- Simplify: f(0) = -0(1)(-2) = 0
So, the y-intercept is at (0, 0).
Step 3: Analyze the end behavior
To analyze the end behavior, we look at the leading term of the equation (-x^2) as x approaches positive and negative infinity.
- As x approaches positive infinity, -x^2 becomes -∞. Therefore, the graph approaches negative infinity as x goes to positive infinity.
- As x approaches negative infinity, -x^2 becomes -∞. Therefore, the graph approaches negative infinity as x goes to negative infinity.
Step 4: Determine the behavior near the x-intercepts
To determine the behavior near the x-intercepts, we analyze the signs of the factors when x is close to each intercept.
- For x < -1, the factors (x+1) and (x-2) are negative, while -x^2 is positive. Therefore, the graph is below the x-axis approaching x = -1 from below.
- For -1 < x < 0, the factor (x+1) is positive, while -x^2 and (x-2) are negative. Therefore, the graph is above the x-axis, crossing it at x = 0 and continuing towards negative infinity.
- For 0 < x < 2, the factors -x^2 and (x-2) are negative, while (x+1) is positive. Therefore, the graph is below the x-axis, crossing it at x = 0 and approaching x = 2 from below.
- For x > 2, all the factors (-x^2, x+1, and x-2) are negative. Therefore, the graph is below the x-axis approaching x = 2 from above.
Step 5: Sketch the graph
Now that we have gathered all the necessary information, we can sketch the graph of f(x) = -x^2(x+1)(x-2) by plotting the x and y-intercepts and drawing the curve based on the behavior identified in step 3 and step 4.
The graph should resemble a downward opening cubic curve passing through the points (0, 0), (-1, 0), and (2, 0) with the ends of the curve pointing towards negative infinity.