how do you sketch these graphs:
k is a positve constant:
y=(x+k)(x-2k)
and
y=x(x-k)(x-5k)
To sketch the graphs of these equations, we can follow a few steps:
1. Identify the x-intercepts: Solve the equation y = 0 to find the values of x where the graph intersects the x-axis. These points are where the function changes sign from positive to negative or vice versa.
2. Find the y-intercept: Substitute x = 0 into the equation and calculate the corresponding value of y. This point represents where the graph intersects the y-axis.
3. Determine the behavior for large values of x: Look at the leading term(s) of the equation to determine the behavior of the graph as x approaches positive or negative infinity. This will help you identify any horizontal or slant asymptotes.
4. Observe the critical points: If there are any critical points (where the derivative is zero or undefined), evaluate the function at those points to determine if they are local maximums or minimums. This will help you understand the shape of the graph between the intercepts.
5. Sketch the graph: Based on the information gathered from steps 1-4, plot the x and y intercepts, any additional critical points, and connect them smoothly to draw the graph.
Let's apply these steps to the given equations:
1. For the equation y = (x + k)(x - 2k):
- The x-intercepts occur when y = 0, so we set (x + k)(x - 2k) = 0:
x + k = 0 or x - 2k = 0
x = -k or x = 2k
- The y-intercept is found by substituting x = 0 into the equation:
y = (0 + k)(0 - 2k) = -2k^2
2. For the equation y = x(x - k)(x - 5k):
- The x-intercepts occur when y = 0, so we set x(x - k)(x - 5k) = 0:
x = 0 or x - k = 0 or x - 5k = 0
- The y-intercept is found by substituting x = 0 into the equation:
y = 0
3. To determine the behavior for large values of x, we look at the leading term(s):
- For the first equation, the leading term is x^2. As x approaches infinity or negative infinity, the graph will open upward.
- For the second equation, the leading term is x^3. As x approaches infinity or negative infinity, the graph will have the same behavior as the leading term, which is determined by the coefficient of x^3 (positive or negative).
4. If there are any critical points in either equation, calculate their values and determine if they are local maximums or minimums. Since the given equations don't have any critical points, we can skip this step.
5. Now, with the information gathered, we can plot the points and sketch the graphs accordingly. The y-intercept for both equations is at (0, 0). The x-intercepts for the first equation are at (-k, 0) and (2k, 0). The x-intercepts for the second equation are at (0, 0), (k, 0), and (5k, 0). Connect these points smoothly to sketch the graphs.
Note: Without knowing the actual values of k, we can't determine the exact shape or position of the graphs, but we can follow the steps above to get a general idea of their behavior.