Investigating Transformations of Quadratics Open Google Chrome. Go to desmos.com, and click the big red button labelled Launch Calculator.

Transcription

1 Investigating Transformations of Quadratics Open Google Chrome. Go to desmos.com, and click the big red button labelled Launch Calculator. Optional: You can create an account or sign into Desmos. This lets you save your graphs for later. The Graph y = ax 2 Compared to y = x 2 1. Graph this relation: y = x 2 Vertex: Axis of symmetry: Maximum or minimum: Optimal value: Zeros/roots: Pro Tip Type shift+6 on your keyboard, the ^ character, to get an exponent. e.g. y=x^2 will become y = x 2 Direction of opening: 2. Graph this relation: y = 2x 2 Explain how the graph of y = 2x 2 is different from the graph of y = x Remove or hide the graph of y = 2x 2 We don t want it in the way for the next part. 4. Graph this relation: y = ax 2 a isn t one of our variables (those are x and y). a is called a parameter. Desmos offers to make you a slider to try different values of the parameter a. Do that: click the blue button to make a slider. Use the slider to try the different values of a. Compare the graphs of y = ax 2 and y = x 2 :

2 a-value a-value in interval notation y = ax 2 compared to y = x 2 a > 1 a (1, + ) a = 1 a [1,1] 0 < a < 1 a (0,1) a = 0 a [0,0] 1 < a < 0 a ( 1,0) a = 1 a [ 1, 1] a < 1 or a (, 1) 5. What doesn t change about the graph when you change the value of a? 6. How can you tell from the value of a which way the graph will open? 7. Which values of a give you a graph that is thinner (rises more quickly) than y = x 2? We say that these graphs have been stretched vertically by a factor of a. 8. Which values of a give you a graph that is wider (rises more slowly) than y = x 2? We say that these graphs have been compressed vertically by a factor of a.

3 Part 2: The Graph y = x 2 + k Compared to y = x 2 1. Graph these relations and describe their attributes: Attribute y = x 2 y = x Vertex Axis of symmetry Maximum or minimum Optimal value Zeros/roots Direction of opening What is different about the graph of y = x compared to y = x 2? 2. Remove or hide the graph of y = x We don t want it in the way for the next part. 3. Graph this relation: y = x 2 + k Make a slider, and use it to try the different values of k. Compare the graphs of y = x 2 and y = x 2 + k: k-value k-value in interval notation y = x 2 + k compared to y = x 2 k = 0 k [0,0] k > 0 k (0, + ) k < 0 k (, 0)

4 4. What doesn t change about the graph when you change the value of k? 5. How can you tell from the value of k which way the graph will move? We say that these graphs have been translated vertically. Part 3: Combining a and k 1. Graph this relation: y = ax 2 + k Use sliders to describe the values of a and k that transform the graph to have the following attributes: Attributes a and k for these attributes Vertex above the x- axis Vertical stretch compared to y = x 2 Vertex below the x-axis Vertical compression compared to y = x 2 No zeros/roots (careful: two ways!) Two roots (careful: two ways!) One root (really two equal roots!)

5 Part 4: The Graph y = (x h) 2 Compared to y = x 2 2. Graph these relations and describe their attributes: Attribute y = x 2 y = (x 3) 2 y = (x [ 3]) 2 Vertex Axis of symmetry Maximum or minimum Optimal value Zeros/roots Direction of opening What is different about the graph of y = (x 3) 2 compared to y = x 2? 3. Remove or hide the graph of y = (x 3) 2 4. Graph this relation: y = (x h) 2 Make a slider, and use it to try the different values of h. Compare the graphs of y = x 2 and y = (x h) 2 : h-value h-value in interval notation y = (x h) 2 compared to y = x 2 h = 0 h [0,0] h > 0 h (0, + ) h < 0 h (, 0)

6 5. What doesn t change about the graph when you change the value of h? 6. How can you tell from the value of h which way the graph will move? Part 5: Putting it all together We say that these graphs have been translated horizontally. 1. Graph this relation: y = a(x h) 2 + k Describe in words how changing each parameter s value (a, h, and k) in y = a(x h) 2 + k will transform (change) the graph of y = x 2. Parameter in y = a(x h) 2 + k How it transforms y = x 2 a h k

7 2. Making Predictions Explain how the graph of y = x 2 is transformed to give the graph of each relation below, and sketch each one on a scrap page. Verify your predictions with Desmos. a) y = 3x 2 b) y = x 2 7 c) y = (x + 8) 2 d) y = 2x e) y = 2 3 x2 f) y = 7 2 (x 1) Describing Transformations Write an equation for each graph below based on the transformations applied to y = x 2. a) Vertical translation of +4 units. b) Horizontal translation of 5 units. c) Horizontal translation of units. d) Vertical stretch by a factor of 3. e) Flipped/reflected vertically (i.e. now opens down). f) Vertical compression by a factor of 1 5. g) Horizontal translation of +3 units, vertical translation of 8 units, vertical compression by a factor of 1, flipped/reflected vertically. 6