Math Quiz 2 Page 8 Spring 2016 Professor Dr M

Math 012quiz 2page 8math 012 Quiz 2spring 2016professor Dr Mary D

Find at least three ordered pairs that satisfy the following equation and graph the line through them. Show all work.

Find at least three ordered pairs that satisfy the following equation and graph the line through them. Show all work.

Find at least five ordered pairs that satisfy the following equation and graph the function through them. Show all work.

Write an equation of a line through the point (2, 4) that is perpendicular to the x-axis. Graph the line on the grid below or create your own graph. State the slope of this line.

Given the linear equation, find the slope and y-intercept of the line. Show all work below.

State the domain and the range of the relation graphed below. Determine whether or not the relation is a function and explain your reasoning.

Given, find each of the following, showing all work: a) b) c) d) e)

Given the points (4, -2) and (-6, 2): a) Find the slope of the line through the points. b) Write an equation in point-slope form of the line through the points. c) Convert the equation to slope-intercept form. d) Convert the equation to standard form. e) Graph the equation. You may use the axes provided, or create your own graph.

a) Write an equation of a vertical line through the point (-3, 5). b) Write an equation of a horizontal line through the point (-7, -2). c) Find the slope of a line parallel to the line with equation 3 x - 7 y = 21. d) Find the slope of a line perpendicular to the line with equation 2 x + 3 y = 5. e) Write an equation in point-slope form of the line through the point (-3, 2) perpendicular to the line with equation 2 x + 3 y = 5.

The number of Burger King restaurants worldwide in 2012 was 35,752. In 2007, there were 34,561 Burger King restaurants worldwide. Let y be the number of Burger King restaurants in the year x, where x = 0 represents the year 2007. a) Write a linear equation that models the growth in the number of Burger King restaurants worldwide in terms of x. [Hint: the line must pass through the points (0, 34561) and (5, 35752)]. b) Use this equation to predict the number of Burger King restaurants worldwide in the year 2014. c) Explain what the slope for this line means in the context of the problem.

End of quiz: please remember to sign and date the statement in the box on the first page of the quiz.

Paper For Above instruction

The given set of problems from the Math 012 quiz covers fundamental concepts of linear equations, graphing, and interpreting data within the context of real-world applications. These problems are designed to evaluate understanding of how to find solutions that satisfy equations, derive equations of lines, interpret slopes and intercepts, analyze functions, and model data with linear equations. This paper will systematically address each problem, demonstrating the approach and reasoning to arrive at solutions, emphasizing clarity and accuracy in algebraic manipulations, graphing, and interpretation.

Problem 1: Finding Ordered Pairs for a Given Equation

Without the specific equation provided, the general approach involves selecting arbitrary values for x or y and solving for the corresponding y or x to find ordered pairs conforming to the equation. For example, if the equation is linear, such as y = 2x + 1, choosing x-values like -1, 0, and 1 would yield y-values of -1, 1, and 3, respectively, giving the ordered pairs (-1, -1), (0, 1), and (1, 3). These points can then be graphed to visualize the line. Similar processes apply for other equations, ensuring to verify that each ordered pair satisfies the original equation. This method allows for multiple solutions and a better understanding of the relationship between x and y.

Problem 2: Finding Ordered Pairs for Another Equation

The same approach applies: assign values to one variable, solve for the other, and verify the solutions. For instance, if the equation is y = -x + 3, choosing x-values of -2, 0, and 2 results in y-values of 5, 3, and 1, respectively, providing the points (-2, 5), (0, 3), and (2, 1). These can be plotted to solidify comprehension of the linear relationship. The process reinforces understanding of how the coefficients influence the slope and intercept of the line, and how to interpret the graph accordingly.

Problem 3: Finding Multiple Ordered Pairs for a Function

Given a quadratic or other nonlinear function, such as y = x^2 - 2, selecting x-values like -2, 0, 1, 2, 3 yields y-values of 2, -2, -1, 2, and 7, respectively. The points (-2, 2), (0, -2), (1, -1), (2, 2), and (3, 7) are plotted to graph the parabola. Showing all work involves calculating each y-value step-by-step, which offers visual insight into the function’s shape and the nature of its solutions. For functions like absolute value or exponential, similar steps are followed with appropriate calculations to generate multiple points.

Problem 4: Equation of a Line Perpendicular to the x-axis

Any line perpendicular to the x-axis is vertical, characterized by a constant x-value. Passing through the point (2, 4), the line’s equation is x = 2. The slope of a vertical line is undefined, which is a critical concept. Graphically, this line is a vertical line crossing the x-axis at x=2 extending infinitely in the y-direction. This illustrates the principle that vertical lines are perpendicular to horizontal lines and have no defined slope, emphasizing the geometric relationships in coordinate geometry.

Problem 5: Slope and Intercept of a Linear Equation

Considering the linear form y = mx + b, where m is the slope and b is the y-intercept, the process involves rewriting the given equation in slope-intercept form. For example, if the provided equation is 2x + 3y = 6, solving for y gives:

3y = -2x + 6


y = (-2/3)x + 2

Thus, the slope (m) is -2/3, and the y-intercept (b) is 2. This approach clarifies the key parameters of the line, with the slope indicating the rate of change and the intercept representing where the line crosses the y-axis. Showing each algebraic step ensures full comprehension and accurate interpretation.

Problem 6: Domain, Range, and Function Determination

The domain consists of all possible x-values for which the relation is defined, while the range involves all y-values. For a graphed relation, the domain can often be read directly from the x-axis extent, and the range from the y-axis extent. For example, if the graph covers x-values from -3 to 3 and y-values from -2 to 4, then the domain is [-3, 3], and the range is [-2, 4]. To determine whether it is a function, verify if each x-value has only one y-value. If so, the relation qualifies as a function; otherwise, it does not. Explaining this involves examining the horizontal line test and the graph’s features.

Problem 7: Calculations of Algebraic Expressions

Since the specific expressions are not provided, typical questions involve simplifying algebraic expressions, evaluating functions at specific values, or performing operations such as addition, subtraction, multiplication, or division of functions. The key is to show each step meticulously to demonstrate understanding and correctness in algebraic manipulations.

Problem 8: Analyzing Two Points and Formulating Line Equations

Given points (4, -2) and (-6, 2), first compute the slope:

m = (2 - (-2)) / (-6 - 4) = (4) / (-10) = -2/5

Point-slope form using point (4, -2):

y - (-2) = -2/5(x - 4)


y + 2 = -2/5(x - 4)

Converting to slope-intercept form:

y = -2/5 x + (8/5) - 2 = -2/5 x - 2/5

Standard form:

2/5 x + y = -2/5


Multiply through by 5:
2x + 5y = -2

Graphing involves plotting the points or using the equation's intercepts and slope.

Problem 9: Lines with Specific Conditions and Slopes

a) Vertical line through (-3, 5): x = -3.

b) Horizontal line through (-7, -2): y = -2.

c) Parallel to 3x - 7y = 21: The slope form is y = (3/7)x - 3, so the slope is 3/7.

d) Perpendicular to 2x + 3y = 5: slope of the given line is -2/3, so the perpendicular slope is 3/2.

e) Equation in point-slope form, through (-3, 2), with slope 3/2:

y - 2 = 3/2(x + 3)

Problem 10: Modeling Growth with a Linear Equation

Points are (0, 34561) for the year 2007, and (5, 35752) for 2012. The slope (rate of increase) is:

m = (35752 - 34561)/ (5 - 0) = 1191/5 = 238.2

Equation in point-slope form:

y - 34561 = 238.2(x - 0)


y = 238.2x + 34561

To predict for 2014 (x=7):

y = 238.2 × 7 + 34561 = 1667.4 + 34561 = 36228.4

The slope indicates the average increase of approximately 238.2 restaurants per year, reflecting steady growth over the period. Understanding this slope contextually helps interpret the expansion rate of Burger King establishments.

Conclusion

This analysis provided detailed methods for solving various algebraic and geometric problems, emphasizing systematic approaches, algebraic manipulation, and contextual interpretation. Mastery of these concepts facilitates understanding linear relationships, their graphical representations, and real-world applications such as business growth models. Such foundational skills are vital for advanced mathematical studies and practical problem-solving.

References

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  • OpenStax. (2021). College Algebra. Rice University. https://openstax.org/details/books/college-algebra
  • Stewart, J. (2016). Calculus: Concepts and Contexts (4th ed.). Brooks Cole.
  • Swokowski, E., & Cole, J. (2018). Algebra and Trigonometry with Analytic Geometry (12th ed.). Cengage.
  • Target Learning. (2019). Linear Functions and Models. https://targetlearning.com/linear-functions
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  • Wiley. (2018). Graphing Linear Equations. https://wiley.com/graphinglinear
  • Zill, D. G. (2018). Algebra and Trigonometry (6th ed.). Cengage Learning.

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