![]() Vertical shifts: The graph of f(x) versus the graph of f(x) + C. ![]() Reflection over the y-axis: The graph of f(x) versus the graph of f(-x). If you would like to review examples on the following, click on Example: Since the point (1, 0) is on the x-axis, the point would not move. , it would be shifted up 22 units to (b, 11). If the point (b, -11) is located on the graph of If we shifted the point (a, 8) down 16 units, it would wind up at (a, - 8) units. For example, the point (a, 8) is located 8 units up from the x-axis. Every point on the graph of would be shifted up or down twice it’s distance from the x-axis. Fold the graph of over the x-axis so that it would be superimposed on the graph of.The graph of is a reflection over the x-axis of the graph of. ![]() What exactly does that mean? Well for one thing, it means if there is a point (a, b) on the graph of, we know that the point (a, - b) is located on the graph of. This means both graphs are symmetric to each other with respect to the x-axis. Note that the graph of the function is superimposed on the graph of the function. Mentally fold the coordinate system at the x-axis.Note that both points have the same x-coordinate and the y-coordinate’s differ by a minus sign.Since neither of the graphs cross the y-axis, there is no y-intercept. The graphs of both functions cross the x-axis at x = 1.This verifies that the domain of both functions is the set of positive real numbers. You can see that the graphs of both functions are located in quadrants I and IV to the right of the y-axis.The domain of both functions is the set of positive real numbers.The graph to the right of the y-axis is the graph of the function, and the graph on the left to the left of the y-axis is the graph of the function. How would you move the graph of so that it would be superimposed on the graph of ? Where would the point (1, 0) on.Describe the relationship between the two graphs.What do these two points have in common?.Find the point (2, f(2)) on the graph of and find ( 2, g( 2)) on the graph of.What is the x-intercept and the y-intercept on the graph of the function ? What is the x-intercept and the y-intercept on the graph of the function ?.In what quadrants is the graph of the function located? In what quadrants is the graph of the function located?.and answer the following questions about each graph: Graph the function and the function on the same rectangular coordinate system. The graph of 3 - g(x) involves the reflection of the graph of g(x) across the x-axis and the upward shift of the reflected graph 3 units. For example, the graph of - f(x) is a reflection of the graph of f(x) across the x-axis. Whenever the minus sign (-) is in front of the function notation, it indicates a reflection across the x-axis. We will also illustrate how you can use graphs to HELP you solve logarithmic problems. In the case of $\,y = 3f(x)\,$ the $\,3\,$ is ‘on the outside’ we're dropping $\,x\,$ in the $\,f\,$ box, getting the corresponding output, and then multiplying by $\,3\.$ This is a vertical stretch.In this section we will illustrate, interpret, and discuss the graphs of logarithmic functions. (that is, transformations that change the $\,y\,$ and transformations involving $\,x\.$ When talking about transformations involving ![]() Transformations Involving $\,y\,$ and $\,x\,$ Moves to a point $\,(ka,b)\,$ on the graph ofĬalled horizontal scaling (stretching/shrinking). This is called a horizontal stretch.Ī point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ Which moves the points farther away from the Should use when you are given the graph of $\,y=f(x)\,$ $\,y\,$ work the way you would expect them to work-they are intuitive. Is found by taking the graph of $\,y=f(x)\,$ Thus, the graph of $\,y=3f(x)\,$ is found by taking the graph of $\,y=f(x)\,$Īre of the form $\,\bigl(x,f(x)\bigr)\.$ These two requests mean exactly the same thing!Īre of the form $\,\bigl(x,3f(x)\bigr)\.$ $\,f\,$ is a picture of all points of the form:Īnd $\,f(x)\,$ is the corresponding output. You should be able to do a problem like this:Ī function is a rule: it takes an input, and gives a unique output. We will explore stretching and shrinking a graph, Makes it easy to graph a wide variety of functions, Stretch or shrink vertically or horizontally.Īn understanding of these transformations ![]() That will change the graph in a variety of ways.įor example, you can move the graph up or down, There are things that you can DO to an equation of the form Graphing Tools: Vertical and Horizontal Translations.Click here for a printable version of the discussion below. ![]()
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