Graph y = x3and do one transformation at a time. Y y 8 8 4 4 x x -4 4 4 Step 3: -4 Step 4: Example: Multiple Transformations Example:Graph using the graph ofy = x3. There are three basic transformations that can be applied to graphs of linear functions: sliding the line around (translation), flipping the line (reflection), and stretching the line. you will see that the graph has been stretched vertically by a factor of 2 and shifted. y = |2x| Example:y = |2x|is the graph of y = |x| shrunk horizontally by 2. The graph of a function is reflected about the y -axis if each x. If 0 < c < 1, the graph of y = f(cx) is the graph of y = f(x) stretchedhorizontally by c. If 0 1, the graph of y = f(cx) is the graph of y = f(x) shrunk horizontally by c. x –4 4 Vertical Stretching and Shrinking Vertical Stretching and Shrinking If c > 1 then the graph of y= cf(x) is the graph of y = f(x) stretched vertically by c. Y 4 is the graphof y = x2shrunk vertically by. A negative a reflects it, and if 01, it vertically stretches the parabola. Example: Shift, Reflection y Example: The graph of y x2+ 3is the graph of y x2shifted upward three units.Then shiftthe graphthree units to the left. The parabola is translated (c,d) units, b reflects across y, but this just reflects it across the axis of symmetry, so it would look the same. Y y 4 4 x x 4 4 – 4 -4 Example: Reflections Example:Graph y =–(x + 3)2using the graph of y = x2. y =f(–x) y = f(x) y =–f(x) The graph of the function y =–f(x) is the graph ofy = f(x)reflected in the x-axis. The graph of the function y = f(–x)is the graphof y = f(x) reflected in the y-axis. y x The graph of a function may be a reflection of the graph of a basic function. Reflection about the x-axis Reflection about the y-axis Vertical shifting or stretching Horizontal shifting or stretching Tell me if Im wrong, but I believe that in any function, you have to do the stretching or the shrinking before the shifting. First make a vertical shift 4 units downward. Y y 4 4 x x -4 -4 (–1, –2) Example: Vertical and Horizontal Shifts Example:Graph the function using the graph of. VERTICAL AND HORIZONTAL SHIFTING: Example 1: Sketch the graph of f(x) x and g(x) x +1. f(x) = x3 g(x) = (x – 2)3 h(x) = (x + 4)3 SHIFTING, STRETCHING AND REFLECTING GRAPHS I. Y 4 4 -4 x Example: Horizontal Shifts Example:Use the graph of f(x) = x3to graphg(x) = (x – 2)3 and h(x) = (x + 4)3. c +c If c is a positive real number, then the graph of f(x + c) is the graph of y = f(x)shifted to the leftc units. There are three basic transformations that can be applied to graphs of linear functions: sliding the line around (translation), flipping the line (reflection), and stretching the line (scaling.
Horizontal Shifts y x Horizontal Shifts If c is a positive real number, then the graph of f(x – c) is the graph of y = f(x)shifted to the rightc units. Y 8 4 x 4 -4 -4 Example: Vertical Shifts Example:Use the graph of f(x) = |x| to graph thefunctions g(x) = |x| +3 and h(x) = |x| –4. If c is a positive real number, the graph of f(x) –cis the graph of y = f(x)shifted downwardc units. Vertical and horizontal shifts in the graph of yf(x) are represented as follows. Vertical Shifts y x VerticalShifts If c is a positive real number, the graph of f(x) + cis the graph of y = f(x)shifted upwardc units. y = x2 + 3 y = x2 The graph of y = –x2is the reflection of the graph of y = x2 in the x-axis. 8 4 x 4 -4 -4 -8 The graphs of many functions are transformations of the graphs of very basic functions. Right now every cycle starts at 0 and ends at \(2 \pi\) but this will not always be the case.- E N D - Presentation TranscriptĢ.5 Shifting, Reflecting, and Stretching GraphsĮxample: Shift, Reflection y Example: The graph of y = x2+ 3is the graph of y = x2shifted upward three units. There are no reflections in these graphs and they all have an amplitude of 1. Note the five important points that separate each quadrant to help to get a clear sense of the graph. Then, a complete sine wave for each one is drawn. To draw these graphs, the new sinusoidal axis for each graph is drawn first. a reflection with respect to the x-axis, draw the resulting graph by. The graphs of the following three functions are shown below: The graph of cf(x) is the graph of f stretched vertically (from the x-axis). Then proceed to graph amplitude and reflection about that axis as opposed to the \(x\) axis. At the start of the problem identify the vertical shift and immediately draw the new sinusoidal axis. Forming a mirror image of a graph across a line is called reflecting the graph across the line. The most straightforward way to think about vertical shift of sinusoidal functions is to focus on the sinusoidal axis, the horizontal line running through the middle of the sine or cosine wave. Here you will see how \(d\) controls the vertical shift. Recall that \(a\) controls amplitude and the \(\pm\) controls reflection. The general form of a sinusoidal function is:
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