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Questions 1) Find the equation of the circle titled "question 1". Write the equation in standard form.

Questions 1) Find the equation of the circle titled "question 1". Write the equation in standard form.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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6 Algebra Questions

1) Find the equation of the circle titled "question 1". Write the equation in standard form.

 

2) Find the domain of the function f(x) = 1/5x + 8  What is the only value of x not in the domain?

 

3) Zhang is planning a ski trip. The equation C = 200 + 23g models the relation between the cost in dollars, C, of the ski trip and the number of skiers, g.

The cost is ____

 

4) Find the value of m and b from the line in the image titled "question 4"

 

5) Find the x- and y-intercepts algebraically: -2x - 3y = -48

 

6) Rewrite using a rational exponent. Assume all variables are positive. (x^72 c^16)^1/8

 

This image shows a graph of a circle plotted on a Cartesian coordinate system. ### Description: - **Grid and Axes**: The graph features a standard Cartesian coordinate system with both horizontal (x-axis) and vertical (y-axis) axes. The gridlines help in accurate plotting of points and interpreting the circle's properties. - **Circle**: A blue circle is centered roughly at the point (-6, 0) on the graph. The circle appears to intersect the x-axis at approximately -11 and -1. - **Axis Labels**: The x-axis ranges from -11 to 1, and the y-axis ranges from -4 to 8. Each unit on the axis is equally spaced and marked clearly. ### Interpretation: - **Center and Radius**: By observing the graph, we can estimate the center of the circle to be at (-6, 0). The circle passes through -1 and -11 on the x-axis, which suggests the radius is approximately 5 units. - **Equations**: If we were to write the equation of this circle, it center (-6, 0) and radius 5 can be expressed as: \[ (x + 6)^2 + y^2 = 25 \] This illustration is useful for understanding concepts in geometry and algebra, particularly those involving the equations of circles, radius, and center transformation, and assists in visualizing mathematical problems in coordinate geometry. ### Educational Usage: - **Geometry Lessons**: This diagram can be used to teach students about the properties of a circle, including how changes in the standard form equation affect the circle's radius and center position. - **Coordinate Geometry**: Perfect for explaining the plotting of geometric shapes on a Cartesian plane, calculating distances, and transforming shapes using algebraic methods.
Transcribed Image Text:This image shows a graph of a circle plotted on a Cartesian coordinate system. ### Description: - **Grid and Axes**: The graph features a standard Cartesian coordinate system with both horizontal (x-axis) and vertical (y-axis) axes. The gridlines help in accurate plotting of points and interpreting the circle's properties. - **Circle**: A blue circle is centered roughly at the point (-6, 0) on the graph. The circle appears to intersect the x-axis at approximately -11 and -1. - **Axis Labels**: The x-axis ranges from -11 to 1, and the y-axis ranges from -4 to 8. Each unit on the axis is equally spaced and marked clearly. ### Interpretation: - **Center and Radius**: By observing the graph, we can estimate the center of the circle to be at (-6, 0). The circle passes through -1 and -11 on the x-axis, which suggests the radius is approximately 5 units. - **Equations**: If we were to write the equation of this circle, it center (-6, 0) and radius 5 can be expressed as: \[ (x + 6)^2 + y^2 = 25 \] This illustration is useful for understanding concepts in geometry and algebra, particularly those involving the equations of circles, radius, and center transformation, and assists in visualizing mathematical problems in coordinate geometry. ### Educational Usage: - **Geometry Lessons**: This diagram can be used to teach students about the properties of a circle, including how changes in the standard form equation affect the circle's radius and center position. - **Coordinate Geometry**: Perfect for explaining the plotting of geometric shapes on a Cartesian plane, calculating distances, and transforming shapes using algebraic methods.
**Understanding Linear Graphs - Slope and Intercepts** This image shows a graph of a linear equation plotted on a coordinate plane. The coordinate plane is marked by horizontal (x-axis) and vertical (y-axis) axes, both ranging from -10 to 10. **Key Features of the Graph:** 1. **Axes**: - The x-axis (horizontal) is labeled from -10 to 10. - The y-axis (vertical) is also labeled from -10 to 10. - Both axes intersect at the origin (0,0). 2. **Grid Lines**: - The grid lines help in reading and plotting points accurately on the graph. Each grid square represents a single unit increment on both the x and y axes. 3. **Line**: - A red line represents the linear equation. - It passes through the points (-2, -10) and (8, 10). - The line extends infinitely in both directions, but within this graph, it starts at point (-10,-10) and ends at (10,10). - The line is straight, indicating a constant slope throughout its length. **Interpreting the Graph:** - **Slope**: The slope (m) of the line is a measure of its steepness. It can be determined by the "rise over run" method, which involves the vertical change (rise) divided by the horizontal change (run) between two points on the line. For example, between the points (-2, -10) and (8, 10), the rise is \( 10 - (-10) = 20 \) and the run is \( 8 - (-2) = 10 \). Thus, the slope \( m = \frac{20}{10} = 2 \). - **Y-Intercept**: The y-intercept (b) is the point where the line crosses the y-axis. In this graph, with the slope provided, the intercept needs to be determined by evaluating the line equation. Substituting a point into the slope-intercept form \( y = mx + b \), for instance, using point (8, 10): \( 10 = 2*8 + b \), which simplifies to \( 10 = 16 + b \). Solving for \( b \), we find \( b = -6 \). Thus, the equation of this
Transcribed Image Text:**Understanding Linear Graphs - Slope and Intercepts** This image shows a graph of a linear equation plotted on a coordinate plane. The coordinate plane is marked by horizontal (x-axis) and vertical (y-axis) axes, both ranging from -10 to 10. **Key Features of the Graph:** 1. **Axes**: - The x-axis (horizontal) is labeled from -10 to 10. - The y-axis (vertical) is also labeled from -10 to 10. - Both axes intersect at the origin (0,0). 2. **Grid Lines**: - The grid lines help in reading and plotting points accurately on the graph. Each grid square represents a single unit increment on both the x and y axes. 3. **Line**: - A red line represents the linear equation. - It passes through the points (-2, -10) and (8, 10). - The line extends infinitely in both directions, but within this graph, it starts at point (-10,-10) and ends at (10,10). - The line is straight, indicating a constant slope throughout its length. **Interpreting the Graph:** - **Slope**: The slope (m) of the line is a measure of its steepness. It can be determined by the "rise over run" method, which involves the vertical change (rise) divided by the horizontal change (run) between two points on the line. For example, between the points (-2, -10) and (8, 10), the rise is \( 10 - (-10) = 20 \) and the run is \( 8 - (-2) = 10 \). Thus, the slope \( m = \frac{20}{10} = 2 \). - **Y-Intercept**: The y-intercept (b) is the point where the line crosses the y-axis. In this graph, with the slope provided, the intercept needs to be determined by evaluating the line equation. Substituting a point into the slope-intercept form \( y = mx + b \), for instance, using point (8, 10): \( 10 = 2*8 + b \), which simplifies to \( 10 = 16 + b \). Solving for \( b \), we find \( b = -6 \). Thus, the equation of this
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