![]() Knowing about even and odd functions is very helpful when studying Fourier Series. There some more examples on this page: Even and Odd Functions Note if we reflect the graph in the x-axis, then the y-axis, we get the same graph. An odd function either passes through the origin (0, 0) or is reflected through the origin.Īn example of an odd function is f( x) = x 3 − 9 x Other important transformations include vertical shifts, horizontal shifts and. This kind of symmetry is called origin symmetry. A math reflection flips a graph over the y-axis, and is of the form y f(-x). This time, if we reflect our function in both the x-axis and y-axis, and if it looks exactly like the original, then we have an odd function. Reflect the graph of f (x) x 1 f ( x) x 1 (a) vertically and (b) horizontally. Thus, for every point of an object, the mirror line is perpendicularly bisects the line segment joining the point with its image. Note if we reflect the graph in the y-axis, we get the same graph (or we could say it "maps onto" itself).Īn odd function has the property f( −x) = −f( x). Example 1: Reflecting a Graph Horizontally and Vertically Reflect the graph of s(t) t s ( t) t (a) vertically and (b) horizontally. The above even function is equivalent to: Reflecting functions are functions whose graphs are reflections of each other. ![]() That is, if we reflect an even function in the y-axis, it will look exactly like the original.Īn example of an even function is f( x) = x 4 − 29 x 2 + 100 Reflections of graphs involve reflecting a graph over a specific line. We say the reflection "maps on to" the original.Īn even function has the property f( −x) = f( x). But sometimes, the reflection is the same as the original graph. We really should mention even and odd functions before leaving this topic.įor each of my examples above, the reflections in either the x- or y-axis produced a graph that was different. Reflection in y-axis (green): f( −x) = −x 3 − 3 x 2 − x − 2 Even and Odd Functions Reflection in x-axis (green): − f( x) = − x 3 + 3 x 2 − x + 2 Figures are usually reflected across either the x x or the y y -axis. In a reflection, the figure flips across a line to make a mirror image of itself. The green line also goes through 2 on the y-axis. You can identify a reflection by the changes in its coordinates. Note that the effect of the "minus" in f( −x) is to reflect the blue original line ( y = 3 x + 2) in the y-axis, and we get the green line, which is ( y = −3 x + 2). ![]() Now, graphing those on the same axes, we have: Now for f(− x)į( −x) = −3 x + 2 (replace every " x" with a " −x"). What we've done is to take every y-value and turn them upside down (this is the effect of the minus out the front). Note that if you reflect the blue graph ( y = 3 x + 2) in the x-axis, you get the green graph ( y = −3 x − 2) (as shown by the red arrows). When we reflect across the y-axis, the image point is the same height, but has the opposite position from left to right. Reflections create mirror images of points, keeping the same distance from the line. When you graph the 2 lines on the same axes, it looks like this: We can plot points after reflecting them across a line, like the x-axis or y-axis. Our new line has negative slope (it goes down as you scan from left to right) and goes through −2 on the y-axis. going uphill as we go left to right) and y-intercept 2. You'll see it is a straight line, slope 3 (which is positive, i.e. If you are not sure what it looks like, you can graph it using this graphing facility. Let's see what this means via an example. If \(00\), represents a vertical stretch if \(|a|>1\) or compression if \(0However, in this example, the minus sign is inside the function, leaving one to intuit that it is the x-values, not the y-values, that are being negated. The left tail of the graph will approach the asymptote \(y=0\), and the right tail will increase without bound. Once these incident rays strike the mirror, reflect them according to the two rules of reflection for concave mirrors. That is why the graph of y f(x) was a reflection of the graph of y f(x) across the x-axis. If \(b>1\),the function is increasing.Example 4.2.2: Graphing a Shift of an Exponential Function. State the domain, (, ), the range, (d, ), and the horizontal asymptote y d. The graph of the function \(f(x)=b^x\) has a y-intercept at \((0, 1)\),domain \((−\infty, \infty)\),range \((0, \infty)\), and horizontal asymptote \(y=0\). See Example. Shift the graph of f(x) bx up d units if d is positive, and down d units if d is negative. ![]() The domain is \((−\infty,\infty)\) the range is \((0,\infty)\) the horizontal asymptote is \(y=0\).
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