How do I sketch a curve with equation y = x(x-2)^2 ?
Expand first the equation. Find the intercepts. Find the domain and range of the function.
The intercepts will serve as the turning points. Since the leading coefficient (of x^3) is positive, the graph of f is up on the right and down on the left and hence the range of f is the set of all real numbers.
Or to make it simple, assign values of x and y and place it in a table. Plot the points in the graph and connect the dots.
the turning points are in between the x-intercepts.
To sketch the curve with the equation y = x(x-2)^2, you can follow these steps:
Step 1: Identify key points and intercepts:
- x-intercepts: These occur when y = 0. Set y = 0 in the equation and solve for x. In this case, set x(x-2)^2 = 0, which gives x = 0, x = 2. Thus, the curve intersects the x-axis at x = 0 and x = 2.
- y-intercept: This occurs when x = 0. Substitute x = 0 in the equation to find the y-intercept. In this case, y = 0(0-2)^2 = 0. So, the curve intersects the y-axis at y = 0.
Step 2: Determine the behavior of the curve:
- As x approaches positive or negative infinity, the curve will also approach infinity. This means that the curve will go upwards on both sides.
Step 3: Sketch the curve:
- Plot the identified intercepts and key points on a coordinate system.
- Based on the identified behavior, draw a smooth curve passing through the intercepts and key points.
- Since the function is quadratic, the curve will be symmetric about the line x = 1 as the equation contains (x-2), which means as x moves away from 1 in both directions, the function remains the same.
- It might be helpful to calculate a few additional points on the curve by substituting various values of x into the equation to get a more accurate representation of the shape.
Overall, the sketch should resemble a curve that starts at the y-intercept at the origin, passes through (0,0), goes downward to an x-intercept at x = 2, and upward as x approaches positive and negative infinity. It might have a U-like shape or a parabolic shape depending on the magnitude of x-values used.