Draw the derivative of these graphs. ### Understanding Turning Points and Concavity through Graphical RepresentationsIn this section, we will explore two graphs that illustrate the concepts of turning points and concavity in mathematical functions. These graphs serve as a visual aid to better understand the behavior of polynomial functions. #### Graph 1: A Quadratic Function**Description:**- **Axes:** The x-axis and y-axis range from -4 to 4.- **Curve:** The graph depicts a simple symmetric curve, indicating a quadratic function, $ y = ax^2 + bx + c $.**Key Characteristics:**1. **Turning Point (Vertex):** - The curve reaches a maximum point at the vertex, which is indicated near the top of the graph. - The vertex is marked with a small red segment. At this point, the slope changes from positive to negative.2. **Concavity:** - The graph is concave down (shaped like an upside-down parabola). - This is indicated by the downward curvature of the graph.#### Graph 2: A Cubic Function**Description:**- **Axes:** The x-axis and y-axis range from -4 to 4.- **Curve:** The graph shows a more complex curve indicating a cubic function, $ y = ax^3 + bx^2 + cx + d $.**Key Characteristics:**1. **Turning Points:** - The function has two turning points, indicated by the red segments. - There is a local minimum point and a local maximum point.2. **Concavity:** - The graph exhibits both concave up and concave down sections. - The first section (left side of the curve) is concave up (shaped like a U). - The second section (right side of the curve) is concave down (shaped like an upside-down U).In summary, these graphs provide a clear visual representation of how quadratic and cubic functions behave in terms of their turning points and concavity. Understanding these characteristics is crucial for analyzing and interpreting polynomial functions.

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Math

Calculus

-4 マナ -4 4 4 -4 -4

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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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

Question

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Draw the derivative of these graphs.

### Understanding Turning Points and Concavity through Graphical Representations

In this section, we will explore two graphs that illustrate the concepts of turning points and concavity in mathematical functions. These graphs serve as a visual aid to better understand the behavior of polynomial functions.

#### Graph 1: A Quadratic Function

**Description:**
- **Axes:** The x-axis and y-axis range from -4 to 4.
- **Curve:** The graph depicts a simple symmetric curve, indicating a quadratic function, $ y = ax^2 + bx + c $.

**Key Characteristics:**
1. **Turning Point (Vertex):**
- The curve reaches a maximum point at the vertex, which is indicated near the top of the graph.
- The vertex is marked with a small red segment. At this point, the slope changes from positive to negative.
2. **Concavity:**
- The graph is concave down (shaped like an upside-down parabola).
- This is indicated by the downward curvature of the graph.

#### Graph 2: A Cubic Function

**Description:**
- **Axes:** The x-axis and y-axis range from -4 to 4.
- **Curve:** The graph shows a more complex curve indicating a cubic function, $ y = ax^3 + bx^2 + cx + d $.

**Key Characteristics:**
1. **Turning Points:**
- The function has two turning points, indicated by the red segments.
- There is a local minimum point and a local maximum point.
2. **Concavity:**
- The graph exhibits both concave up and concave down sections.
- The first section (left side of the curve) is concave up (shaped like a U).
- The second section (right side of the curve) is concave down (shaped like an upside-down U).

In summary, these graphs provide a clear visual representation of how quadratic and cubic functions behave in terms of their turning points and concavity. Understanding these characteristics is crucial for analyzing and interpreting polynomial functions.

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