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Open Adobe Illustrator. Switch to the "Tracing" workspace. Add the JPG image that you want to convert to your artboard. Select the image on your artboard. Check the "Preview" box in the Image Trace panel. Try one of the presets in the Image Trace panel. Use the color slider to adjust color complexity. Expand the "Advanced" section of the Image Trace panel. Use the "Paths" slider to adjust how close the path follows the pixels. Use the "Corners" slider to adjust how rounded your corners will be. Use the "Noise" slider to reduce bleeds. Click "Trace" when you're satisfied. Click the "Expand" button. Export the image as a vector file.
Adobe Illustrator is a professional image creation program, and is the easiest way to create vector images from JPG files. If you don't have access to Illustrator, see the next section for instructions on using the free GIMP and Inkscape utilities. You can do this by clicking the menu in the upper-right corner and selecting "Tracing." You'll see the "Image Trace" panel appear. You can do this from the File menu or by dragging and dropping the file. The tracing options in the Image Trace panel will become active. This will allow you to see what the different settings will do before applying them, but it will increase the time between making changes. There are five preset buttons along the top of the panel, and more presets available in the drop-down menu. The top row of buttons include the following presets:  Auto Color - Creates a stylized set of colors based on the original colors. High Color - Attempts to recreate all of the original colors. Low Color - Creates a simplified version of the original colors. Grayscale - Replaces colors with shades of gray. Black and White - Reduces colors to black and white. Images converted to vectors don't usually show well with all of their natural colors, so you'll get a better result by reducing the number of colors used. This will lead to a "flatter" look for the image. This will reveal more detailed controls for the trace. Moving the slider to the left makes the paths looser, while sliding it to the right makes the paths tighter. A looser path will lead to smoother edges. Moving the slider to the left will make the corners rounder, which will lead to a smoother image. The Noise slider determines what groupings of pixels are considered "noise" and not included in the trace. This can help straighten lines and smooth out rough spots. Illustrator will then perform the tracing action, which may take a few moments to complete. This will convert your tracing object into actual vector paths, and will replace your JPG image with the vector. Once you're finished with the tracing, you can export the finished image as a vector file.  Click the File or Illustrator menu and select "Save As." Save a copy as an .ai file. This will let you easily re-open the file in Illustrator and make more edits. Select a vector format from the "Save As Type" menu. These include SVG (webpage) and PDF (print). Do not save the file as PNG or JPG, as these are not vector formats.