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Which graph could be used to find the solution(s) to the system of equations y = log₂(x + 2) and y=x²³ - 2x² -5x + 6? 10 -0 3 7 8 9 10 9 10 J O 1. -10 -D -8 -10 -9 --8 47 * --$ -3 4 -10 10 * S → 4 A -10

Which graph could be used to find the solution(s) to the system of equations y = log₂(x + 2) and y=x²³ - 2x² -5x + 6? 10 -0 3 7 8 9 10 9 10 J O 1. -10 -D -8 -10 -9 --8 47 * --$ -3 4 -10 10 * S → 4 A -10

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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### Understanding the Intersection Points of Two Functions #### Diagram 1: Graph of y = f(x) and y = g(x) In the first diagram, we have two functions plotted on a coordinate axis. - **Horizontal and Vertical Axes:** The horizontal axis is labeled as \( x \) ranging approximately from -10 to 10, and the vertical axis is labeled as \( y \) ranging from approximately -20 to 20. - **Blue Curve:** Represents the function \( f(x) \). This function starts around \( y = 0 \) when \( x = -10 \), increases gradually and appears to stabilize, approaching an asymptote around \( y = 6 \) as \( x \) approaches positive infinity. - **Orange Curve:** Represents the function \( g(x) \). This function starts around \( y \approx -10 \) at \( x = -10 \), passes through the x-axis around \( x = -3 \), increases to a peak around \( y = 10 \), and decreases steeply crossing the x-axis at multiple points to negative values around \( x = 5 \), and again increases sharply. **Points of Intersection:** The curves intersect at approximately four points: 1. \( x \approx -2 \) 2. \( x \approx 0.5 \) 3. \( x \approx 2 \) 4. \( x \approx 8 \) #### Diagram 2: Graph of y = h(x) and y = g(x) In the second diagram, we have a different function \( h(x) \) plotted along with the same \( g(x) \) function from the first diagram: - **Horizontal and Vertical Axes:** Same range as the first diagram, with \( x \) from about -10 to 10, and \( y \) from about -20 to 20. - **Blue Curve:** Represents the function \( h(x) \). This function behaves similarly to \( f(x) \) from the first diagram. - **Orange Curve:** Represents the same \( g(x) \) function as in the first diagram. **Points of Intersection:** The curves intersect at approximately three points: 1. \( x \approx -7 \) 2. \( x \approx 0.2 \) 3. \( x \approx 6 \) Understanding the behavior and intersections of these functions is key to solving equations involving
Transcribed Image Text:### Understanding the Intersection Points of Two Functions #### Diagram 1: Graph of y = f(x) and y = g(x) In the first diagram, we have two functions plotted on a coordinate axis. - **Horizontal and Vertical Axes:** The horizontal axis is labeled as \( x \) ranging approximately from -10 to 10, and the vertical axis is labeled as \( y \) ranging from approximately -20 to 20. - **Blue Curve:** Represents the function \( f(x) \). This function starts around \( y = 0 \) when \( x = -10 \), increases gradually and appears to stabilize, approaching an asymptote around \( y = 6 \) as \( x \) approaches positive infinity. - **Orange Curve:** Represents the function \( g(x) \). This function starts around \( y \approx -10 \) at \( x = -10 \), passes through the x-axis around \( x = -3 \), increases to a peak around \( y = 10 \), and decreases steeply crossing the x-axis at multiple points to negative values around \( x = 5 \), and again increases sharply. **Points of Intersection:** The curves intersect at approximately four points: 1. \( x \approx -2 \) 2. \( x \approx 0.5 \) 3. \( x \approx 2 \) 4. \( x \approx 8 \) #### Diagram 2: Graph of y = h(x) and y = g(x) In the second diagram, we have a different function \( h(x) \) plotted along with the same \( g(x) \) function from the first diagram: - **Horizontal and Vertical Axes:** Same range as the first diagram, with \( x \) from about -10 to 10, and \( y \) from about -20 to 20. - **Blue Curve:** Represents the function \( h(x) \). This function behaves similarly to \( f(x) \) from the first diagram. - **Orange Curve:** Represents the same \( g(x) \) function as in the first diagram. **Points of Intersection:** The curves intersect at approximately three points: 1. \( x \approx -7 \) 2. \( x \approx 0.2 \) 3. \( x \approx 6 \) Understanding the behavior and intersections of these functions is key to solving equations involving
**Finding the Solution(s) to a System of Equations** Which graph could be used to find the solution(s) to the system of equations \( y = \log_2(x + 2) \) and \( y = x^3 - 2x^2 - 5x + 6 \)? **Graph 1:** - **Type:** Cartesian graph - **Axes:** - Horizontal axis: \( x \) ranging from -10 to 10 - Vertical axis: \( y \) ranging from -10 to 10 - **Functions:** - The blue curve represents the function \( y = \log_2(x + 2) \). This function indicates a logarithmic growth, starting near negative infinity as \( x \) approaches -2 and increasing without bound. - The orange curve represents the function \( y = x^3 - 2x^2 - 5x + 6 \). This is a cubic polynomial, displaying characteristic cubic behavior with multiple turning points and potentially multiple roots (where the curve crosses the x-axis). - **Intersections:** - The points where the blue curve intersects the orange curve indicate the solution(s) to the system of equations. These are the \( x \)-values for which both functions yield the same \( y \)-value. **Graph 2:** - **Type:** Cartesian graph - **Axes:** - Horizontal axis: \( x \) ranging from -10 to 10 - Vertical axis: \( y \) ranging from -10 to 10 - **Functions:** - The blue curve represents the function \( y = \log_2(x + 2) \). Similar to Graph 1, this function starts near negative infinity as \( x \) approaches -2 and increases without bound. - The orange curve represents another instance of the cubic function \( y = x^3 - 2x^2 - 5x + 6 \). It also displays typical cubic behavior with multiple turning points. - **Intersections:** - The points of intersection between the blue and orange curves in this graph also represent the solution(s) to the system of equations \( y = \log_2(x + 2) \) and \( y = x^3 - 2x^2 - 5x + 6 \). --- To determine which graph correctly represents the system
Transcribed Image Text:**Finding the Solution(s) to a System of Equations** Which graph could be used to find the solution(s) to the system of equations \( y = \log_2(x + 2) \) and \( y = x^3 - 2x^2 - 5x + 6 \)? **Graph 1:** - **Type:** Cartesian graph - **Axes:** - Horizontal axis: \( x \) ranging from -10 to 10 - Vertical axis: \( y \) ranging from -10 to 10 - **Functions:** - The blue curve represents the function \( y = \log_2(x + 2) \). This function indicates a logarithmic growth, starting near negative infinity as \( x \) approaches -2 and increasing without bound. - The orange curve represents the function \( y = x^3 - 2x^2 - 5x + 6 \). This is a cubic polynomial, displaying characteristic cubic behavior with multiple turning points and potentially multiple roots (where the curve crosses the x-axis). - **Intersections:** - The points where the blue curve intersects the orange curve indicate the solution(s) to the system of equations. These are the \( x \)-values for which both functions yield the same \( y \)-value. **Graph 2:** - **Type:** Cartesian graph - **Axes:** - Horizontal axis: \( x \) ranging from -10 to 10 - Vertical axis: \( y \) ranging from -10 to 10 - **Functions:** - The blue curve represents the function \( y = \log_2(x + 2) \). Similar to Graph 1, this function starts near negative infinity as \( x \) approaches -2 and increases without bound. - The orange curve represents another instance of the cubic function \( y = x^3 - 2x^2 - 5x + 6 \). It also displays typical cubic behavior with multiple turning points. - **Intersections:** - The points of intersection between the blue and orange curves in this graph also represent the solution(s) to the system of equations \( y = \log_2(x + 2) \) and \( y = x^3 - 2x^2 - 5x + 6 \). --- To determine which graph correctly represents the system
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