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 they multiplied inside the function by something larger than 1, or outside the function by something smaller than 1, then they did a function compression.įor instance, looking at y = x 2 − 4, you can see that multiplying outside the function doesn't change the location of the zeroes, but multiplying inside the function does.In this section we will illustrate, interpret, and discuss the graphs of logarithmic functions. If they multiplied inside the function by something smaller than 1, or outside the function by something larger than 1, then they did a function stretch.If these points line up horizontally, then they multiplied inside the function. If these points line up vertically, then they multiplied outside of the function. If the y-intercepts remain the same, then they multiplied inside the function.
If the x-intercepts remain the same, then they multiplied outside of the function. To figure out whether a graph is stretched or compressed, in comparison with the original graph, look at the max/min points and the x- and y-intercepts. How can you tell if a graph is stretched or compressed? Sometimes, though, it helps to look at the zeroes of the graph (if it has more than one) or turning points (that is, the max/min points). In my experience, it can feel just about impossible to discern whether a multiplication transformation involved multiplying inside or outside the function. The graphic below shows g( x) in blue, g(2 x) in dark green, and g(½ x) in dark red: Instead of multiplying by 2 and by ½ on the outside, I'll instead multiply on the inside that is, I'll multiply on the x that is the argument of the function.įor clarity, the two new functions are these: On the other hand, we can also multiply inside a function. This behavior - namely, that the x-intercepts don't move and the max/min points line up vertically - is the hallmark of multiplying a function on the outside by some number. All of the x-intercepts are the same, and the max/min points line up vertically (that is, the max/min points occur at the same x values, but at different y-value heights). And multiplying by ½ (which is smaller than 1) caused the highs and lows of the original graph to contract, drawing closer to the x-axis. Using the function g( x) = (2 − x)( x + 1)( x + 4), the graphic below shows g( x) in blue, 2 g( x) in dark green, and ½ g( x) in dark red:Īs you can see, multiplying on the outside of the function by 2 (which is larger than 1) caused the highs and lows of the original graph to go higher and lower. This fixedness of the intercepts makes sense, since x-intercepts are where the function is zero, and zero times the number is still gonna be zero. But - and this is important - the x-intercepts will remain the same. If you multiply a function by a number, then the function will get taller (if the number is greater than 1) or shorter (if the number is less than 1). The more important thing is to understand the difference between multiplying outside the function and multiplying inside the function.) What happens if you multiply a function by a number? (The terms "squeezed", "shortened", "compressed", "stretche", etc, seem not to have fixed meanings in this context, which makes this type of function transformation even more confusing. If the graph of the new function is shorter or narrower than the original, then the function has been compressed. If the graph of the new function is taller or wider than the original function's graph, then the function has been stretched.
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