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. If \(b If \(0 If \(0 1\), then the graph will be compressed by \(1/b\).If \(a > 1\), then the graph will be stretched.Given a function \(f(x)\), a new function \(g(x) = a f(x)\), where \(a\) is a constant, is a vertical stretch or a vertical compression of the function \(f(x)\). The graph of an odd function is symmetric about the origin. A function called an odd function if for every input \(x\) \(f(x)= -f(-x)\). The graph of an even function is symmetric about the y-axis. Even and Odd Functions (Symmetry)Ī function is called an even function if for every input \(x\) \(f(x) = f( -x)\). Given a function \(f(x)\), a new function \(g(x) = f(-x)\) is a horizontal reflection of the function \(f(x)\) over the y-axis.įigure 3 You will see the horizontal and vertical reflection of a function \(f(x)\). Given a function \(f(x)\), a new function \(g(x) = - f(x)\) is a vertical reflection of the function \(f(x)\) over the x-axis. When \(k\) is negative, then the graph will translate \(k\) units downward on the y-axis.įigure 2 There is Vertical Shift of the function \(f(x) = ∛x\) thus \(k = 1\). When \(k\) is positive, then the graph will translate \(k\) units upward on the y- axis. Given the function \(f(x)\), a new function \(g(x) = f(x) + k\), where \(k\) is a constant of the function \(f(x)\). When \(h\) is positive, then the graph \(y = f(x)\) will translate h units to the right of the x- axis.When \(h\) is negative, then the graph \(y = f(x)\) will translate \(h\) units to the left of the x-axis.įigure 1 There is Horizontal Shift of the function \(f(x) = ∛x\) thus \(h = +1\) Note: that the function is shifting left after the addition of 1. Given a function \(f\), a new function \(g(x) = f(x – h)\), where \(h\) is a constant of the function \(f\). The transformations vary as follows: Horizontal Translation
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