Use the vertex and intercepts to graph the quadratic function f (x) = -x² + 2x + 3.
Use the vertex and intercepts to graph the quadratic function f (x) = -x² + 2x + 3.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Title: Graphing Quadratic Functions Using Vertex and Intercepts**
**Objective:**
Learn how to graph a quadratic function using its vertex and intercepts.
**Function:**
\[ f(x) = -x^2 + 2x + 3 \]
**Instructions:**
To graph the quadratic function, follow these steps:
1. **Find the Vertex:**
- The vertex of a quadratic function \( ax^2 + bx + c \) is given by:
\[ x = -\frac{b}{2a} \]
- Substitute \( x \) back into the function to find the \( y \)-coordinate of the vertex.
2. **Identify Intercepts:**
- **Y-intercept:** Set \( x = 0 \) and find \( f(x) \).
- **X-intercepts:** Solve \( f(x) = 0 \) for \( x \).
3. **Plot the Points on a Graph:**
- Use the vertex and intercepts to sketch the parabola.
- Note that the parabola opens downward because the coefficient of \( x^2 \) is negative.
**Graph Analysis:**
The accompanying graph provides a coordinate plane with labeled axes \( x \) and \( y \). Use this grid to accurately plot the vertex and intercepts.
**Summary:**
Learning to graph quadratic functions by identifying key points like the vertex and intercepts helps in understanding the function’s behavior and its real-world applications.
---
**Note:** Make sure to use a calculator or algebraic method to accurately find the vertex and intercepts for precise plotting.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F719f3c35-5b56-4005-8b10-ea1cf51ff043%2F0ae0378a-873c-473b-9404-732d90a80704%2Fsb9ei9n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Graphing Quadratic Functions Using Vertex and Intercepts**
**Objective:**
Learn how to graph a quadratic function using its vertex and intercepts.
**Function:**
\[ f(x) = -x^2 + 2x + 3 \]
**Instructions:**
To graph the quadratic function, follow these steps:
1. **Find the Vertex:**
- The vertex of a quadratic function \( ax^2 + bx + c \) is given by:
\[ x = -\frac{b}{2a} \]
- Substitute \( x \) back into the function to find the \( y \)-coordinate of the vertex.
2. **Identify Intercepts:**
- **Y-intercept:** Set \( x = 0 \) and find \( f(x) \).
- **X-intercepts:** Solve \( f(x) = 0 \) for \( x \).
3. **Plot the Points on a Graph:**
- Use the vertex and intercepts to sketch the parabola.
- Note that the parabola opens downward because the coefficient of \( x^2 \) is negative.
**Graph Analysis:**
The accompanying graph provides a coordinate plane with labeled axes \( x \) and \( y \). Use this grid to accurately plot the vertex and intercepts.
**Summary:**
Learning to graph quadratic functions by identifying key points like the vertex and intercepts helps in understanding the function’s behavior and its real-world applications.
---
**Note:** Make sure to use a calculator or algebraic method to accurately find the vertex and intercepts for precise plotting.
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