There is a curve with equation y=3/x, xis not equal 0. Sketch the curve with equation y=3/(x+2), xis not equal to -2, showing the coordinates of any point at which the curve crosses a coordinate axis.
It is just the graph of y = 3/x, shifted left by two.
Plug in x=0 and y=0 to find the intercept(s).
By now, you should know that y=3/x has a horizontal asymptote at y=0.
No matter how big x gets, y is never zero!
To sketch the curve with equation y = 3/(x+2), we can start by examining the original curve equation y = 3/x and observe its behavior.
1. When x > 0, the value of y is positive because the numerator (3) is positive.
2. When x < 0, the value of y is negative because the numerator (3) is positive and the denominator (x) is negative.
3. As x approaches 0, y tends toward infinity because the denominator (x) approaches 0.
Using this information, we can proceed with sketching the curve:
1. First, plot some key points on the original curve y = 3/x. Choose x-values of -3, -2, -1, 1, 2, and 3, and find the corresponding y-values by substituting them into the equation.
For example:
- When x = -3, y = 3/(-3) = -1
- When x = -2, y = 3/(-2) = -1.5
- When x = -1, y = 3/(-1) = -3
- When x = 1, y = 3/(1) = 3
- When x = 2, y = 3/(2) = 1.5
- When x = 3, y = 3/(3) = 1
2. Plot these points on a graph paper, taking care to label them with their respective coordinates.
3. Now, consider the transformed equation y = 3/(x+2). Since we have shifted the curve two units to the left, all the x-values need to be decreased by 2.
For example:
- When x = -1, the new x-value is -1 - 2 = -3
- When x = 0, the new x-value is 0 - 2 = -2
- When x = 1, the new x-value is 1 - 2 = -1
- When x = 3, the new x-value is 3 - 2 = 1
- and so on...
4. Plot the new points on the graph paper, using the transformed x-values and the y-values from step 1. Label these points as well.
5. Connect the points smoothly, maintaining the same characteristic behavior, i.e., positive slope when x > 0, negative slope when x < 0, and approaching infinity as x approaches 0.
6. Lastly, mark the points where the curve intersects the x and y-axes. These occur when either x = 0 or y = 0.
By following these steps, you will have successfully sketched the curve with equation y = 3/(x+2) and identified the coordinate points where the curve crosses the x and y-axes.