graph the following function using transformations. Be sure to graph all the stages on one graph. state the domain and range.
y= -3/x/+1
NOT QUITE SURE HOW TO MAKE THE STRAIGHT UP AND DOWN LINE, INSTEAD OF THIS--/.
This graph is a v shape with vertex at (0,0) with a ray in the first quad with slope 1 and a ray in the second quad with slope -1
The +1 sends this graph up one
It does not change the curve.
The -flip the v downward
The 3 changes the curve. Instead of the slope begin +/- 1, it is now +/- 3
Hope this will help you
To graph the function y = -3/x + 1 using transformations, we can follow these steps:
Step 1: Start with the parent function y = 1/x.
The parent function is a hyperbola and has a vertical asymptote at x = 0. It has a horizontal asymptote at y = 0, meaning the graph approaches but never reaches the x-axis or y-axis.
Step 2: Reflect the parent function about the x-axis.
To reflect the function about the x-axis, multiply the function by -1. This will change the shape of the graph, but it will still have the same asymptotes.
Step 3: Stretch the function horizontally.
To stretch the function horizontally, multiply the x-values by a constant. Since we want to make it a straight up and down line, we can set this constant equal to 1. This step does not change the asymptotes.
Step 4: Shift the function upwards by 1 unit.
To shift the function upwards by 1 unit, add 1 to the y-values. This will move the entire graph above the x-axis, shifting the horizontal asymptote from y = 0 to y = 1.
Now let's graph the function on one graph:
1. Determine the domain and range:
The domain of the function y = -3/x + 1 is all real numbers except x = 0 since there is a vertical asymptote at x = 0. So, the domain is (-∞, 0) U (0, +∞).
The range of the function is all real numbers except y = 1 since there is a horizontal asymptote at y = 1. So, the range is (-∞, 1) U (1, +∞).
2. Plot the asymptotes:
Draw a vertical dashed line at x = 0 (vertical asymptote) and a horizontal dashed line at y = 1 (horizontal asymptote).
3. Plot a few points:
Choose various x-values and find their corresponding y-values using the equation y = -3/x + 1. As an example, let's choose x = -3, -2, -1, 1, 2, and 3.
For x = -3, y = -3/(-3) + 1 = -1 + 1 = 0. Plot the point (-3, 0).
For x = -2, y = -3/(-2) + 1 = 1.5 + 1 = 2.5. Plot the point (-2, 2.5).
For x = -1, y = -3/(-1) + 1 = 3 + 1 = 4. Plot the point (-1, 4).
For x = 1, y = -3/(1) + 1 = -3 + 1 = -2. Plot the point (1, -2).
For x = 2, y = -3/(2) + 1 = -1.5 + 1 = -0.5. Plot the point (2, -0.5).
For x = 3, y = -3/(3) + 1 = -1 + 1 = 0. Plot the point (3, 0).
4. Draw the graph:
Connect the points with a smooth curve that approaches the asymptotes. Remember, the graph should be a reflection of the parent function y = 1/x, stretched horizontally, and shifted upwards by 1 unit.
The final graph should be a hyperbola reflected about the x-axis, with a vertical asymptote at x = 0, a horizontal asymptote at y = 1, and passing through the points (-3, 0), (-2, 2.5), (-1, 4), (1, -2), (2, -0.5), and (3, 0).
Make sure to label the axes and indicate the asymptotes.
Hope this helps!