horizontal stretch log function

To obtain the graph of: y = f(x) + c: shift the graph of y= f(x) up by c units y = f(x) - c: shift the graph of y= f(x) down by c units y = f(x - c): shift the graph of y= f(x) to the right by c units y = f(x + c): shift the graph of y= f(x) to the left by c units Example:The graph below depicts g(x) = ln(x) and a function, f(x), that is the result of a transformation on ln(x). As we mentioned in the beginning of the section, transformations of logarithmic functions behave similar to those of other parent functions. Consider the following base functions, (1) f (x) = x 2 - 3, (2) g(x) = cos (x). State the domain, range, and asymptote.Since the function is [latex]f\left(x\right)=2{\mathrm{log}}_{4}\left(x\right)[/latex], we will note thatThis means we will stretch the function [latex]f\left(x\right)={\mathrm{log}}_{4}\left(x\right)[/latex] by a factor of 2.Consider the three key points from the parent function, [latex]\left(\frac{1}{4},-1\right)[/latex], [latex]\left(1,0\right)[/latex], and [latex]\left(4,1\right)[/latex].Label the points [latex]\left(\frac{1}{4},-2\right)[/latex], [latex]\left(1,0\right)[/latex], and [latex]\left(4,\text{2}\right)[/latex].The domain is [latex]\left(0,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is x = 0.Sketch a graph of [latex]f\left(x\right)=\frac{1}{2}{\mathrm{log}}_{4}\left(x\right)[/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.The domain is [latex]\left(0,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is Sketch the graph of [latex]f\left(x\right)=5\mathrm{log}\left(x+2\right)[/latex]. Round to the nearest thousandth.Solve [latex]5\mathrm{log}\left(x+2\right)=4-\mathrm{log}\left(x\right)[/latex] graphically. State the domain, range, and asymptote.Since the function is [latex]f\left(x\right)={\mathrm{log}}_{3}\left(x\right)-2[/latex], we notice This means we will shift the function [latex]f\left(x\right)={\mathrm{log}}_{3}\left(x\right)[/latex] down 2 units.Consider the three key points from the parent function, [latex]\left(\frac{1}{3},-1\right)[/latex], [latex]\left(1,0\right)[/latex], and [latex]\left(3,1\right)[/latex].The new coordinates are found by subtracting 2 from the Label the points [latex]\left(\frac{1}{3},-3\right)[/latex], [latex]\left(1,-2\right)[/latex], and [latex]\left(3,-1\right)[/latex].The domain is [latex]\left(0,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is x = 0.Sketch a graph of [latex]f\left(x\right)={\mathrm{log}}_{2}\left(x\right)+2[/latex] alongside its parent function. State the domain, range, and asymptote.Since the function is [latex]f\left(x\right)={\mathrm{log}}_{3}\left(x - 2\right)[/latex], we notice [latex]x+\left(-2\right)=x - 2[/latex].The vertical asymptote is [latex]x=-\left(-2\right)[/latex] or Consider the three key points from the parent function: [latex]\left(\frac{1}{3},-1\right)[/latex], [latex]\left(1,0\right)[/latex], and [latex]\left(3,1\right)[/latex].Plot and label the points [latex]\left(\frac{7}{3},-1\right)[/latex], [latex]\left(3,0\right)[/latex], and [latex]\left(5,1\right)[/latex].The domain is [latex]\left(2,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is Sketch a graph of [latex]f\left(x\right)={\mathrm{log}}_{3}\left(x+4\right)[/latex] alongside its parent function. Include the key points and asymptote on the graph. Suppose c > 0. The lesson Graphing Tools: Vertical and Horizontal Scaling in the Algebra II curriculum gives a thorough discussion of horizontal and vertical stretching and shrinking. A horizontal stretch or shrink by a factor of 1/k means that the point (x, y) on the graph of f(x) is transformed to the point (x/k, y) on the graph of g(x). Include the key points and asymptote on the graph. Include the key points and asymptote on the graph. First, we move the graph left 2 units and then stretch the function vertically by a factor of 5. Round to the nearest thousandth.Now that we have worked with each type of transformation for the logarithmic function, we can summarize each in the table below to arrive at the general equation for transforming exponential functions.All transformations of the parent logarithmic function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] have the form[latex] f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d[/latex]where the parent function, [latex]y={\mathrm{log}}_{b}\left(x\right),b>1[/latex], isFor [latex]f\left(x\right)=\mathrm{log}\left(-x\right)[/latex], the graph of the parent function is reflected about the What is the vertical asymptote of [latex]f\left(x\right)=-2{\mathrm{log}}_{3}\left(x+4\right)+5[/latex]?The coefficient, the base, and the upward translation do not affect the asymptote. State the domain, range, and asymptote.The domain is [latex]\left(2,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is When the parent function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] is multiplied by –1, the result is a The function [latex]f\left(x\right)={\mathrm{-log}}_{b}\left(x\right)[/latex]The function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(-x\right)[/latex]The graphs below summarize the key characteristics of reflecting [latex]f(x) = \log_{b}{x}[/latex] horizontally and vertically.Sketch a graph of [latex]f\left(x\right)=\mathrm{log}\left(-x\right)[/latex] alongside its parent function. For example, look at the graph in the previous example. State the domain, range, and asymptote.The domain is [latex]\left(-4,\infty \right)[/latex], the range [latex]\left(-\infty ,\infty \right)[/latex], and the asymptote Sketch a graph of [latex]f\left(x\right)={\mathrm{log}}_{3}\left(x\right)-2[/latex] alongside its parent function. Where k=the horizontal stretch/compression; if k<0, the functions has undergone a horizontal reflection across the y-axis. Include the key points and asymptotes on the graph. Function Transformations: Horizontal and Vertical Stretch and Compression This video explains to graph graph horizontal and vertical stretches and compressions in the form af(b(x-c))+d.