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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Question
![**Title: Curve Sketching of Quadratic Functions**
**Instructions:**
Use the steps to curve sketching to graph the following functions.
**Problem 3:**
\[ f(x) = x^2 - 12x + 36 \]
**Graph Explanation:**
This function is a quadratic expression, represented by a parabola. The key features you should identify and graph are:
1. **Vertex**: The vertex form of a quadratic function is helpful to identify the vertex directly. For this function, it can be transformed as such: \( f(x) = (x - 6)^2 \), indicating the vertex is at point (6, 0).
2. **Axis of Symmetry**: The axis of symmetry for this function is the vertical line \( x = 6 \).
3. **Direction**: Since the coefficient of \( x^2 \) is positive, the parabola opens upwards.
4. **Y-intercept**: When \( x = 0 \), \( f(x) = 36 \). Thus, the y-intercept is at (0, 36).
5. **X-intercepts**: The points where the graph intersects the x-axis can be found by setting \( f(x) = 0 \). Solving gives a double root at \( x = 6 \), indicating this is also the vertex.
This curve sketching approach helps in understanding the behavior of the quadratic function and plotting it accurately on a coordinate plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f9e4cd1-015a-4a96-a677-2fe562598422%2F3954c807-e53e-4d32-8f82-567381a68daf%2F36c3n0e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Curve Sketching of Quadratic Functions**
**Instructions:**
Use the steps to curve sketching to graph the following functions.
**Problem 3:**
\[ f(x) = x^2 - 12x + 36 \]
**Graph Explanation:**
This function is a quadratic expression, represented by a parabola. The key features you should identify and graph are:
1. **Vertex**: The vertex form of a quadratic function is helpful to identify the vertex directly. For this function, it can be transformed as such: \( f(x) = (x - 6)^2 \), indicating the vertex is at point (6, 0).
2. **Axis of Symmetry**: The axis of symmetry for this function is the vertical line \( x = 6 \).
3. **Direction**: Since the coefficient of \( x^2 \) is positive, the parabola opens upwards.
4. **Y-intercept**: When \( x = 0 \), \( f(x) = 36 \). Thus, the y-intercept is at (0, 36).
5. **X-intercepts**: The points where the graph intersects the x-axis can be found by setting \( f(x) = 0 \). Solving gives a double root at \( x = 6 \), indicating this is also the vertex.
This curve sketching approach helps in understanding the behavior of the quadratic function and plotting it accurately on a coordinate plane.
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