College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter2: Functions And Graphs
Section2.4: Writing And Graphing Equations Of Lines
Problem 104E: Applications Waste management The corrosive waste is industrial sewage limits the useful life of the...
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![**Title: Solving Equations Using Graphs**
**Introduction:**
In this lesson, we will explore how graphical representations of functions can help us solve equations. Specifically, we will be using the graphs of two quadratic functions to find their intersection points, which correspond to the solutions of a given equation.
**The Given Functions:**
We have two functions:
\[ f(x) = -\frac{1}{2}(x-6)^2 + 5 \]
\[ g(x) = \frac{1}{4}x^2 - 4 \]
These functions are plotted on the provided coordinate plane.
**Graph Analysis:**
The graph shows two quadratic curves:
1. The function \( f(x) = -\frac{1}{2}(x-6)^2 + 5 \) represents a downward-opening parabola, with its vertex located at the point (6, 5).
2. The function \( g(x) = \frac{1}{4}x^2 - 4 \) represents an upward-opening parabola, with its vertex at the point (0, -4).
**Using the Graphs to Solve the Equation:**
A) To solve the equation \( \frac{1}{4}x^2 - 4 = -\frac{1}{2}(x-6)^2 + 5 \), we need to determine where the graphs of the two functions \( f(x) \) and \( g(x) \) intersect. The x-coordinates of these intersection points are the solutions to the equation, because at these points, \( f(x) \) and \( g(x) \) have the same value.
**Visual Representation:**
The plotted graphs are depicted on a coordinate plane with the y-axis ranging approximately from -8 to 8 and the x-axis ranging from -8 to 12. The intersection points of the two curves are visually observable.
**Finding Solutions:**
B) From the graph, we can observe the coordinates of the intersection points, which are approximately around \( x = 2 \) and \( x = 8 \). Thus, the solutions to the equation are the x-values at the points where the two graphs intersect, specifically in this case, x ≈ 2 and x ≈ 8.
**Conclusion:**
By using the graph of the functions \( f(x) \) and \( g(x) \), we can visually determine the solutions to the equation](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F740f5a09-bcb7-4356-922d-836c20cd171d%2F859bacdf-f81a-46c0-8649-aca14bf4a8c8%2Ftjio7zo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving Equations Using Graphs**
**Introduction:**
In this lesson, we will explore how graphical representations of functions can help us solve equations. Specifically, we will be using the graphs of two quadratic functions to find their intersection points, which correspond to the solutions of a given equation.
**The Given Functions:**
We have two functions:
\[ f(x) = -\frac{1}{2}(x-6)^2 + 5 \]
\[ g(x) = \frac{1}{4}x^2 - 4 \]
These functions are plotted on the provided coordinate plane.
**Graph Analysis:**
The graph shows two quadratic curves:
1. The function \( f(x) = -\frac{1}{2}(x-6)^2 + 5 \) represents a downward-opening parabola, with its vertex located at the point (6, 5).
2. The function \( g(x) = \frac{1}{4}x^2 - 4 \) represents an upward-opening parabola, with its vertex at the point (0, -4).
**Using the Graphs to Solve the Equation:**
A) To solve the equation \( \frac{1}{4}x^2 - 4 = -\frac{1}{2}(x-6)^2 + 5 \), we need to determine where the graphs of the two functions \( f(x) \) and \( g(x) \) intersect. The x-coordinates of these intersection points are the solutions to the equation, because at these points, \( f(x) \) and \( g(x) \) have the same value.
**Visual Representation:**
The plotted graphs are depicted on a coordinate plane with the y-axis ranging approximately from -8 to 8 and the x-axis ranging from -8 to 12. The intersection points of the two curves are visually observable.
**Finding Solutions:**
B) From the graph, we can observe the coordinates of the intersection points, which are approximately around \( x = 2 \) and \( x = 8 \). Thus, the solutions to the equation are the x-values at the points where the two graphs intersect, specifically in this case, x ≈ 2 and x ≈ 8.
**Conclusion:**
By using the graph of the functions \( f(x) \) and \( g(x) \), we can visually determine the solutions to the equation
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