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Two students are working to sketch the graph of the function k(x)=1.5(x−2)2+4 .  Both begin by imagining a horizontal translation of the graph of y=x2 .  However they proceed differently and notice they get different results.  Their work is shown below.  To help them understand who is correct, define the function rules for each graph the students drew. h(x)= r(x)= p(x)=

Two students are working to sketch the graph of the function k(x)=1.5(x−2)2+4 .  Both begin by imagining a horizontal translation of the graph of y=x2 .  However they proceed differently and notice they get different results.  Their work is shown below.  To help them understand who is correct, define the function rules for each graph the students drew. h(x)= r(x)= p(x)=

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Two students are working to sketch the graph of the function k(x)=1.5(x−2)2+4 .  Both begin by imagining a horizontal translation of the graph of y=x2 .  However they proceed differently and notice they get different results.  Their work is shown below.  To help them understand who is correct, define the function rules for each graph the students drew.

h(x)=

r(x)=

p(x)=

The image displays two sequences of transformations applied to quadratic functions, one belonging to "Carly" and the other to "Susie." ### Carly's Sequence of Transformations 1. **First Graph (y = f(x)):** - The blue parabola opens upwards. - The vertex is at (2, 1). 2. **Second Graph (y = g(x)):** - The new blue parabola remains opening upwards. - The vertex has moved to (3, 4). - This indicates a translation of the vertex right by 1 unit and up by 3 units. 3. **Third Graph (y = h(x)):** - The blue parabola is still upward facing. - The vertex is now located at (4, 7). - This suggests another translation right by 1 unit and up by 3 units. ### Susie's Sequence of Transformations 1. **First Graph (y = s(x)):** - The red parabola opens upwards. - The vertex is at (2, 2). 2. **Second Graph (y = r(x)):** - The red parabola continues to open upwards. - The vertex has shifted to (3, 5). - This indicates a translation of the vertex right by 1 unit and up by 3 units. 3. **Third Graph (y = p(x)):** - The red parabola is still facing upwards. - The vertex is located at (4, 8). - This suggests another translation right by 1 unit and up by 3 units. ### Summary Both sequences of transformations illustrate the consistent pattern of translating the vertex of each quadratic parabola to the right by 1 unit and up by 3 units with each transformation step. The transformations maintain the shape and orientation of the parabolas while changing their position on the coordinate plane.
Transcribed Image Text:The image displays two sequences of transformations applied to quadratic functions, one belonging to "Carly" and the other to "Susie." ### Carly's Sequence of Transformations 1. **First Graph (y = f(x)):** - The blue parabola opens upwards. - The vertex is at (2, 1). 2. **Second Graph (y = g(x)):** - The new blue parabola remains opening upwards. - The vertex has moved to (3, 4). - This indicates a translation of the vertex right by 1 unit and up by 3 units. 3. **Third Graph (y = h(x)):** - The blue parabola is still upward facing. - The vertex is now located at (4, 7). - This suggests another translation right by 1 unit and up by 3 units. ### Susie's Sequence of Transformations 1. **First Graph (y = s(x)):** - The red parabola opens upwards. - The vertex is at (2, 2). 2. **Second Graph (y = r(x)):** - The red parabola continues to open upwards. - The vertex has shifted to (3, 5). - This indicates a translation of the vertex right by 1 unit and up by 3 units. 3. **Third Graph (y = p(x)):** - The red parabola is still facing upwards. - The vertex is located at (4, 8). - This suggests another translation right by 1 unit and up by 3 units. ### Summary Both sequences of transformations illustrate the consistent pattern of translating the vertex of each quadratic parabola to the right by 1 unit and up by 3 units with each transformation step. The transformations maintain the shape and orientation of the parabolas while changing their position on the coordinate plane.
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