Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Here are nine of the basic graph types. F of X is the square root of X. The domain includes zero and goes out to Infinity. The range includes zero and goes out to Infinity. Here's a rough sketch of what it looks like. It's neither even or odd because it's not symmetric around the origin or the Y axis. If we look at F of X equal the cube root of X, the domain is all real numbers, which is written negative Infinity to Infinity. The range is negative Infinity to Infinity. It is an odd function because the Y value at negative X equals the opposite of the Y value at X. Here's another sketch of that graph. F of X equals the absolute value of X. The domain is all reals. The range is only the positives and 0. It's an even function because it's symmetric around the Y axis, IE the Y value at negative X is the same thing as the Y value at X. If we look at the constant function F of X equal B, the domain is all reals. The range is just the value for B because no matter what we put in, we're only getting B out. It is an even function because the Y value at X is equivalent to the Y value at negative X. Here's a graph. This is assuming B is whatever that value up is. F of X equal X is called the identity function. The domain is all reals. The range is also all reals. It is an odd function because the Y value at negative X equals the opposite of the Y value at X. Here's a graph. F of X equal X ^2. Domain is all reals. The range is 0 to Infinity, including zero. It's an even function. It's a graph of a parabola, and the Y value at X equals the Y value at negative XF of X equal X ^3. The domain is all reals. The range is all reals. It's an odd function, IE the Y value at negative X equals the opposite value of the Y value at X. Here's a rough sketch. F of X equal 1 / X, sometimes referred to as a reciprocal function. The domain is negative Infinity to 0, union 0 to Infinity. It can't include zero because we don't know how to divide by zero. 1 / 0 is undefined. The range is actually the same thing, negative Infinity to 0, not including zero, union 0 to Infinity. It's an odd function, so F of negative X equals the opposite of F of X. Here's a rough graph of what it would look like. The last one is a linear function. F of X equal MX plus BB is the Y intercept. It's actually a .0 BY intercept and the M is the slope. In the picture to the right I have a slope that is positive and B intercept anywhere we want to have it. Domain is negative Infinity to Infinity range also negative Infinity to Infinity. The function might be odd if b = 0. If B doesn't equal 0, then it's not even or odd transformations. Y equal F of X + K is a vertical shift. If K is greater than 0, we're going to go up. If K is less than 0, we're going to go down. So we're taking your original function and adding some constant to it. The next one, Y equal F of the quantity X + H That's a horizontal shift. If H is greater than 0, it's going to go left. If H is less than 0, it's going to go right. The way I have students think of it sometimes is to think about this inside parenthesis and setting it equal to 0. So if X + H equaled 0, then X would equal negative H and that would tell us how we're shifting left or right. Y equal the opposite of F of X. That's a reflection about the X axis. So whatever the original Y value is, we're now taking the opposite value. Hence the reflection Y equal F of negative X. That's going to be a reflection about the Y axis. If you have Y equaling a * F of X, that's a vertical stretcher shrink. Where a is greater than 1 is a stretch, 0 less than a less than 1 is a shrink. Y equal F of BX is a horizontal stretch or shrink. If zero is less than B is less than one, it's a stretch. If B is greater than one, it's a shrink. One method of graphing these the order to do would be reflections. First stretches and shrinks and then shifts. Now we can do it in lots of different order, but that's usually the easiest to do. Reflections, stretches and shrinks and then shifts. Thank you and have a wonderful day.