To sketch the graph of the equation y^2=(x-2)^2(x-1), we first need to analyze the equation to understand its important characteristics. The equation can be rewritten as y = ±√((x-2)^2(x-1)), suggesting that it is a function that has two branches or parts, as indicated by the ± symbol.
Next, let's identify the key points on the graph:
- At x=1, the graph will intersect the x-axis.
- At x=2, the graph has a point of inflection.
Now, to plot the graph, start by drawing the x-axis and the y-axis. Due to the square root involved in the equation, both positive and negative y values should be considered. Begin by determining the shape and direction of each part of the graph by evaluating the behavior around the key points mentioned above.
- As x approaches 1 on the left side, the value under the square root (x-1) will be negative, resulting in imaginary output. Thus, the graph will start at (1,0) as a vertical tangent.
- As x approaches 1 on the right side, the value under the square root becomes positive, causing the graph to rise up from the x-axis.
- Around x=2, the graph approaches a point of inflection, indicating a change in concavity.
With this information, start plotting the graph by sketching it through the points in a way that reflects the characteristics mentioned earlier. Remember to consider both positive and negative branches when drawing the curve. As you plot more points, you will get a clearer picture of the graph's shape and direction.
explain to approach the following question:
Sketch the graph of y^2=(x-2)^2(x-1)
3 answers
consider the graph of y=(x-2)^2(x-1)
We know it has a double root at x=2, so it is tangent to the graph there
It as a root at x=1, so it crosses there.
That means that we have
y<0 for x in [-∞,1)
y=0 at x=2
y>0 for x in [1,2]U[2,∞)
So since we want y^2=(x-2)^2(x-1)
hat means y is undefined for x<1 and is shaped roughly like the graph of y=(x-2)^2(x-1) everywhere else, just squeezed down toward the x-axis
We know it has a double root at x=2, so it is tangent to the graph there
It as a root at x=1, so it crosses there.
That means that we have
y<0 for x in [-∞,1)
y=0 at x=2
y>0 for x in [1,2]U[2,∞)
So since we want y^2=(x-2)^2(x-1)
hat means y is undefined for x<1 and is shaped roughly like the graph of y=(x-2)^2(x-1) everywhere else, just squeezed down toward the x-axis
From the analysis provided, we can infer that the graph of y^2=(x-2)^2(x-1) will share some similarities with the graph of y=(x-2)^2(x-1) but will be distorted due to the squaring of y. Here is how to approach sketching the graph based on the information given:
1. Begin by considering the original function y=(x-2)^2(x-1). From the discussion, we know that this function has a double root at x=2 and a single root at x=1. It crosses the x-axis at x=1 and is tangent to the x-axis at x=2.
2. Analyze the sign of y in the intervals determined: y<0 for x in (-∞,1), y=0 at x=2, and y>0 for x in [1,2]U[2,∞).
3. Since we are dealing with y^2=(x-2)^2(x-1), we know that y can be both positive and negative for the same x-values, but when squared, it will always be positive. Therefore, the graph of y^2=(x-2)^2(x-1) will have a similar shape to y=(x-2)^2(x-1) but will be "squeezed down" or confined closer to the x-axis.
4. Plot the graph of y=(x-2)^2(x-1) and modify it to ensure that y^2=(x-2)^2(x-1) remains positive for all x-values by reflecting any negative branches above the x-axis.
By following these steps, you will be able to sketch the graph of y^2=(x-2)^2(x-1) with the understanding that it closely resembles the graph of y=(x-2)^2(x-1) but with a vertical stretching effect due to the squaring of the y-values.
1. Begin by considering the original function y=(x-2)^2(x-1). From the discussion, we know that this function has a double root at x=2 and a single root at x=1. It crosses the x-axis at x=1 and is tangent to the x-axis at x=2.
2. Analyze the sign of y in the intervals determined: y<0 for x in (-∞,1), y=0 at x=2, and y>0 for x in [1,2]U[2,∞).
3. Since we are dealing with y^2=(x-2)^2(x-1), we know that y can be both positive and negative for the same x-values, but when squared, it will always be positive. Therefore, the graph of y^2=(x-2)^2(x-1) will have a similar shape to y=(x-2)^2(x-1) but will be "squeezed down" or confined closer to the x-axis.
4. Plot the graph of y=(x-2)^2(x-1) and modify it to ensure that y^2=(x-2)^2(x-1) remains positive for all x-values by reflecting any negative branches above the x-axis.
By following these steps, you will be able to sketch the graph of y^2=(x-2)^2(x-1) with the understanding that it closely resembles the graph of y=(x-2)^2(x-1) but with a vertical stretching effect due to the squaring of the y-values.
1 answer