The document provides step-by-step instructions for performing four transformations on a graph: stretching vertically, stretching horizontally, translating vertically, and translating horizontally. It then discusses drawing the absolute value and reciprocal of the transformed graph. The transformations are applied to the original function g(x) = 2(x-1)2 - 4 to arrive at the final functions h(x) and 1/2(x-1)2 - 4.

1. Transformations Question 2 Solution

2. S o l u t i o n t o ( a ) :

3. To get the graph you want, it is recommended you follow these steps: 1. Stretches (Vertically) 2. Stretches (Horizontally) 3. Translations (Vertically) 4. Translations (Horizontally) The order of these don't matter. The order of these don't matter. This is your original graph and the first transformation we'll do are the stretches of the y-axis. A table of values will be used to show the effect each transformation will have on the graph. x y 3 3 2 1 0 2 1 9 4 9 4 1 0 1 x 2 - - - x y 1 2 3 8 4 12 16 20 20 16 12 8 4 - - - - - 1 - 2 - 3 - 4 -4 x 2 = f(x)

4. 1. Stretches: Vertically (y-axis) 2 (x-1) 2 -4 When you deal with stretches vertically, it doesn't not shift or effect the x-axis in any way. Because if the coefficient in front of the equation, , we multiply all the y-values by 2. As you see on the table below when the stretch is applied, the y-values are doubled, and the x values on the x-axis are unchanged. 2 (x-1) 2 -4 x y 3 3 2 1 0 2 1 18 8 18 8 2 0 2 2x 2 - - - x y 3 3 2 1 0 2 1 9 4 9 4 1 0 1 x 2 - - -

5. x 2 = f(x) y = 2x 2 This is what the graph looks like when the values on the y-axis are doubled. It also makes the graph look 'skinnier'. x y 1 2 3 8 4 12 16 20 20 16 12 8 4 - - - - - 1 - 2 - 3 - 4 -4

6. x y 4 3 2 1 -1 0 18 8 18 8 2 0 2 2(x- 1 ) 2 -2 x y 3 3 2 1 0 2 1 9 4 9 4 1 0 1 x 2 - - - 2. Translations (Horizontally - x-axis) When the graph is being shifted, it does not change the shape of the graph, just moves it. The y-values are not effected by the shift. The graph moves right (when value is positive) or left (when value is negative). When reading the value of the shift off the equation, you read the opposite of the value shown. So, instead of the graph moving -1 units to the left, it is moving 1 unit to the right. 2(x- 1 ) 2 -4

7. This is what the graph looks like with the shift added on to it. x 2 = f(x) y = 2x 2 y = 2(x-1) 2 x y 1 2 3 8 4 12 16 20 20 16 12 8 4 - - - - - 1 - 2 - 3 - -4 4

8. 3. Translation's (Vertically - y-axis) 2(x-1) 2 -4 x y 4 3 2 1 -1 0 14 4 14 4 -2 -4 -2 2(x-1) 2 -4 -2 x y 3 3 2 1 0 2 1 9 4 9 4 1 0 1 x 2 - - - Just like shifting the graph horizontally, when you shift vertically, it does not change the shape of the graph. But it does change the y-values by moving the graph up or down . This number also determines the y-intercept , where x = 0.

9. x 2 = f(x) y = 2x 2 y = 2(x-1) 2 g(x) = 2(x-1) 2 -4 x y 1 2 3 8 4 12 16 20 20 16 12 8 4 - - - - - 1 - 2 - 3 - -4 4

10. g(x) = 2(x-1) 2 -4 This is the graph g(x) = 2(x-1) 2 -4 x y 1 2 3 8 4 12 16 20 20 16 12 8 4 - - - - - 1 - 2 - 3 - -4 4

11. S o l u t i o n t o ( b ) :

12. Absolute Value Graphs An absolute value graph is a graph where you take the negative y-values and flip them over the x-axis. All the negative y-values become positive.Therefore -y becomes y . Absolute value is written like this : |x| Verbally, that would mean, &quot;the absolute value of...&quot; f(x) = |x| = x if x > 0 -x if x < 0 {

13. -Your first step to drawing an absolute value graph is to draw the original graph. In this case, that graph is g(x). Please refer to slide 10 to see our original graph. -Secondly, you take all the points under the x-axis and reflect them over the x-axis (please refer to the next slide). So that for all points: -y --------------> y Then bam you have the absolute value of g(x) which is h(x) on the graph.

14. g(x) = 2(x-1) 2 -4 Absolute Value Graphs h(x) = Ι 2(x-1) 2 -4 Ι x y 1 2 3 8 4 12 16 20 20 16 12 8 4 - - - - - 1 - 2 - 3 - -4 4

15. S o l u t i o n t o ( c ) :

16. Drawing reciprocal graphs The reciprocal of x is: As a sequence gets smaller, the reciprocal gets bigger. As a sequence gets bigger, the reciprocal gets smaller. Invariant points are points that don't change during a transformation. In the case of reciprocal graphs, those points are where the original graph has points at y = -1 and y = 1. (green points on the graph on slide 18) An asymptote is a line on the graph that our function will get closer and closer to, but never touch. Asymptotes are found at where y = 0. 1 x

17. Step one: Draw the original graph. Step two: Find your invariant points (at y = 1 and y = -1) These points won't change on your reciprocal graph. Step three: Find the asymptote(s) at the point(s) where y = 0. (blue dashed lines on our graph) Step four: It's time to actually sketch out our graph. Remember, our points don't change at the invaiant points. That is y = 1 and y = -1. Where the x values get bigger, they get smaller on the reciprocal graph. Where the x values get smaller, they get bigger on the reciprocal graph. Draw these lines toward the asymptotes, but make sure they never touch them (blues solid lines).

18. g(x) = 2(x-1) 2 -4 x y 1 2 3 8 4 12 16 20 20 16 12 8 4 - - - - - 1 - 2 - 3 - -4 4 1 2(x-1) 2 -4 = y