Lesson 6-5: Transforms of Graphs of Functions

Transcription

1 There s an old saying that says a picture is worth a thousand words. I d like to propose a slight modification: a picture is worth a thousand numbers. Anyone who plays with data looks for a way to visualize their data. Scientists, researchers, engineers, economists, business owners, marketing people (this list goes on and on) all collect and analyze data. Often times they feel like they re drowning in numbers as they try to make sense out of them. Oh for a way to picture all these numbers what do they look like? How do they fit together? How do we picture numbers? Well, one way is by graphing them. In chapter 2 we played with graphs of linear equations including the absolute value equation. If you recall, we explored what slight changes to the basic absolute value equation y = x looked like on the graph. Do you remember what we called these simple changes? We called them transformations. Why don t you take a minute and pull out your notes on lesson 2-6 and review what we discovered. In this lesson we will be extending those discoveries to simple quadratic functions. It is very important that you understand how these work. These simple graph transformations work for the graph of almost any function. If you get these down now, it will really help you later on in the course! These techniques will allow you to graph simple functions extremely quickly! Simple graphing transformations reviewed The simple graphing transformations are really quite simple! There are three basic things you can do to the function: 1. Shift it up or down 2. Shift it left or right 3. Flip it up-side-down (ok, this is actually called a reflection, not a translation but ) Today we ll be playing with simple or basic quadratic and absolute value functions. The basic quadratic function f(x) = x 2 is on the left and the basic absolute value function f(x) = x is on the right: Page 1 of 6

2 Quick graphs the up/over trick There is a sneaky little trick you can use to graph a simple quadratics and absolute value functions. First, if you look at the graphs above, you will note they are symmetric about the y-axis the right side is reflected on the left. Second, they both follow a pattern. It is easiest to see with the absolute value graph the V shape is very regular. To get from one point to the next on the right side, you go up one, right one. On the left side it is up one left. In general up one over one. So, plot the vertex (the bend or elbow) then go up one over one on each side for a couple points. Connect the dots and you re done! Now, look at the quadratic on the left. It is very similar but curved. The vertex is at (0, 0). Now, go up one, right one to the point (1, 1). Next go up three, over one to the point (4, 2). Then go up five, over one to the point (9, 3). See the pattern? Go up in by the odd numbers in order and over one each time: up 1 (over 1), up 3 (over 1), up 5 (over 1), up 7 over 1, etc. Do the same on the left side and you have an extremely accurate quick graph! I call this the up/over trick. The transformations Now that you can very quickly graph a quadratic or absolute value function, let s toss in the transformations. To do this, we need to be able to agree on what we ll call the parts of the function. On the left is the basic general form for a simple quadratic function and on the right is the basic general form for a simple absolute value function. The parts we need to discuss are the color coded coefficients: f ( x) a( x b) 2 c g() x a xb c They look very similar. Using these, the transformation rules are: 1. Shift it up or down (the number outside the parenthesis/brackets) Up by c units: add c Down by c units: subtract c 2. Shift it left or right (the number inside the parenthesis/brackets) Left by b units: add b Right by b units: subtract b 3. Flip it up-side-down (the number before the parenthesis/brackets) Open up: a is positive Open down: a is negative Page 2 of 6

3 Shifting left or right Take a look at the transformations above. Does one of them seem backward? Shifting up or down makes complete sense we re moving the graph up or down in the y direction. Up is positive so if we add c it makes sense the graph moves up. Flipping makes sense. Opening up means opening in the positive y direction so the reflection rule saying that a being positive opens up makes sense too! But what about the left or right shift? Positive x on the x-axis means moving to the right. But the transformation rule says that moves the graph to the left! Why? It feels backwards! First, you just need to memorize this rule. Even if it doesn t make sense, memorize it! Now I know it helps if it makes sense, so I ll try to explain it. If it still doesn t make sense, work with it for a bit. If it simply baffles you, no worries, just make sure you memorize the rule! Why is the left/right shift backwards? Consider these two examples: 1. Let s play with the function f(x) = (x + 3) 2. What is the value of x when y is 0? Make sure you read that carefully! 0 = (x + 3) 2 The only number that works is -3. It will give you zero inside the parentheses. 2. With the non-transformed function f(x) = x 2 what is the value of x when y is 0? 0 = x 2 The only number that works is 0! Looking at those two examples, if I add 3 inside the parentheses/brackets, my y value is actually the same as that of the basic non-transformed version when x is 0! The effect is shifting the graph to the left, not the right. Again, even if that doesn t make sense to you, memorize the rule that adding a number inside the parentheses/bracket shifts the graph to the left! Adding a number inside the parentheses shifts the graph to the left! Got it? Page 3 of 6

4 Finding the vertex oh so simple! To graph the function, we need to know where the vertex is. Well the transformation rules tell you exactly where the vertex is! Where is the vertex of the basic functions f ( x) 2 x and g( x) x? Both are at (0, 0). All the transformations do is shift the vertex up/down and left/right! Once you plot the vertex, decide if it opens up or down (the reflection rule). Then plot the graph using the up/over graphing trick! The function quick graphing cookbook recipe! 1. Plot the vertex. Use the up/down and left/right transformations. 2. Open up or down? Use the reflection rule: open up if a is positive. 3. Plot using the up/over graphing trick. Page 4 of 6

5 Practice makes perfect! 2 Let s try this out. Graph the function f ( x) ( x 3) 2. Follow the steps! 1. Plot the vertex: Adding 2 outside: shift up 2 Subtracting 3 inside: shift right 3 (remember this seems backwards!) The vertex will be at (3, 2) 2. Open up or down? The number before the parenthesis is positive: opening up 3. Plot using the up/over graphing trick. Quadratic: up 1, over 1 up 3, over 1 up 5 over 1 each side. Page 5 of 6

6 Another! Graph the function g( x) x 4 1. Remember to follow the steps! 1. Plot the vertex: Subtracting outside: shift down 1 Adding 4 inside: shift left 4 (remember this seems backwards!) The vertex will be at (-4, -1) 2. Open up or down? The number before the brackets is negative: opening down 3. Plot using the up/over graphing trick. Absolute value opening down: down 1, over 1 down 1, over 1 Page 6 of 6