Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Concept explainers
Transformation of Graphs
The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.
Exponential Functions
The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.
Question
![**Sketch a Graph of \( f \):**
Given the function:
\[ f(x) = (x - 4)^2 - 8 \]
1. **Equation Explanation:**
- This is a quadratic function in vertex form, \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.
- For this function, \( h = 4 \) and \( k = -8 \), so the vertex is at \( (4, -8) \).
2. **Vertex Form:**
- The vertex form of a quadratic function emphasizes the vertex, making it easier to graph the parabola by shifting it from the origin.
3. **Transformation:**
- The graph of \( f(x) = (x - 4)^2 - 8 \) is a standard parabola \( y = x^2 \) shifted 4 units to the right and 8 units down.
**Using the Graphing Tool:**
To visually understand the function, use the below graphing method:
1. **Step-by-Step Graphing:**
- **Vertex**: Plot the vertex at \( (4, -8) \).
- **Axis of Symmetry**: The parabola is symmetric about the vertical line \( x = 4 \).
- **Direction of Opening**: Because the coefficient of \( (x - 4)^2 \) is positive (i.e., the \( a \)-value is 1), the parabola opens upwards.
- **Additional Points**: Choose various \( x \)-values, substitute into the function to find corresponding \( y \)-values, and plot these points to shape the parabola accurately.
2. **Graph Illustration:**
- The graph provided has a standard x-y coordinate system with both axes marked from -10 to 10.
- Grid lines are visible to help plot points precisely.
**Click to Enlarge Graph:**
- The tool offers an option to enlarge the graph, making it easier to visualize details and ensure accuracy when plotting points.
By following these steps, students can create an accurate graph of the given quadratic function, enhancing their understanding of vertex form and transformations in quadratic equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff7bd99ef-7092-4177-9c19-951c55d49391%2F00208f8c-c27b-47cd-8a8a-b6d9b161a7c1%2Fb0pe5ja_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Sketch a Graph of \( f \):**
Given the function:
\[ f(x) = (x - 4)^2 - 8 \]
1. **Equation Explanation:**
- This is a quadratic function in vertex form, \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.
- For this function, \( h = 4 \) and \( k = -8 \), so the vertex is at \( (4, -8) \).
2. **Vertex Form:**
- The vertex form of a quadratic function emphasizes the vertex, making it easier to graph the parabola by shifting it from the origin.
3. **Transformation:**
- The graph of \( f(x) = (x - 4)^2 - 8 \) is a standard parabola \( y = x^2 \) shifted 4 units to the right and 8 units down.
**Using the Graphing Tool:**
To visually understand the function, use the below graphing method:
1. **Step-by-Step Graphing:**
- **Vertex**: Plot the vertex at \( (4, -8) \).
- **Axis of Symmetry**: The parabola is symmetric about the vertical line \( x = 4 \).
- **Direction of Opening**: Because the coefficient of \( (x - 4)^2 \) is positive (i.e., the \( a \)-value is 1), the parabola opens upwards.
- **Additional Points**: Choose various \( x \)-values, substitute into the function to find corresponding \( y \)-values, and plot these points to shape the parabola accurately.
2. **Graph Illustration:**
- The graph provided has a standard x-y coordinate system with both axes marked from -10 to 10.
- Grid lines are visible to help plot points precisely.
**Click to Enlarge Graph:**
- The tool offers an option to enlarge the graph, making it easier to visualize details and ensure accuracy when plotting points.
By following these steps, students can create an accurate graph of the given quadratic function, enhancing their understanding of vertex form and transformations in quadratic equations.
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