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Questions and Answers
What are some concepts covered in the chapter 'Preparation for Calculus'?
What are some concepts covered in the chapter 'Preparation for Calculus'?
Sketching the graphs of equations and functions, graphing equations of lines, evaluating and graphing functions, fitting mathematical models to data
Which French mathematician revolutionized the study of mathematics by joining algebra and geometry with the coordinate plane?
Which French mathematician revolutionized the study of mathematics by joining algebra and geometry with the coordinate plane?
What is the main advantage of representing points in the plane by pairs of real numbers?
What is the main advantage of representing points in the plane by pairs of real numbers?
It allows for analytic formulation of geometric concepts and graphical representation of algebraic concepts.
Graphical representation of equations shows the exact and complete graph of the equation.
Graphical representation of equations shows the exact and complete graph of the equation.
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The point (2, 1) is a solution point of the equation 3x - y = 7 because when 2 is substituted for x and 1 is substituted for y, the equation is ________.
The point (2, 1) is a solution point of the equation 3x - y = 7 because when 2 is substituted for x and 1 is substituted for y, the equation is ________.
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What is the mathematical model of the form y = at^2 - bt + c for the CPI data provided?
What is the mathematical model of the form y = at^2 - bt + c for the CPI data provided?
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What is the predicted CPI for the year 2010 based on the model?
What is the predicted CPI for the year 2010 based on the model?
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What is the mathematical model of the form y = at^2 - bt + c for the cellular phone subscriber data provided?
What is the mathematical model of the form y = at^2 - bt + c for the cellular phone subscriber data provided?
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What is the predicted number of cellular phone subscribers for the year 2010 based on the model?
What is the predicted number of cellular phone subscribers for the year 2010 based on the model?
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What is the term used for the points at which the graph intersects the x-axis or y-axis?
What is the term used for the points at which the graph intersects the x-axis or y-axis?
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Which type of symmetry involves the graph being a mirror image of the portion to the left of the y-axis to the right of the y-axis?
Which type of symmetry involves the graph being a mirror image of the portion to the left of the y-axis to the right of the y-axis?
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An equation has an x-intercept when the value of y is zero.
An equation has an x-intercept when the value of y is zero.
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To find the x-intercepts of a graph, let y be ______ and solve the equation for x.
To find the x-intercepts of a graph, let y be ______ and solve the equation for x.
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What is the condition for a quadratic function to have two x-intercepts?
What is the condition for a quadratic function to have two x-intercepts?
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What is the equation for the break-even point?
What is the equation for the break-even point?
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If b^2 - 4ac ≤ 0 and a ≠ 0, the graph of a quadratic function will have only one x-intercept.
If b^2 - 4ac ≤ 0 and a ≠ 0, the graph of a quadratic function will have only one x-intercept.
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The resistance of 1000 feet of solid copper wire at 77°F can be approximated by the model y = ____.
The resistance of 1000 feet of solid copper wire at 77°F can be approximated by the model y = ____.
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What is the formula to calculate the slope of a line passing through two points?
What is the formula to calculate the slope of a line passing through two points?
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What is a point of intersection of the graphs of two equations?
What is a point of intersection of the graphs of two equations?
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How can you find the points of intersection of two graphs?
How can you find the points of intersection of two graphs?
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What are the two points of intersection for x^2 - y = 3 and x - y = 1?
What are the two points of intersection for x^2 - y = 3 and x - y = 1?
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How can you check the points of intersection in Example 5?
How can you check the points of intersection in Example 5?
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What are the two conflicting goals when developing a mathematical model for real-life applications?
What are the two conflicting goals when developing a mathematical model for real-life applications?
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What was the prediction for the carbon dioxide concentration in Earth's atmosphere in the year 2035 according to the quadratic model?
What was the prediction for the carbon dioxide concentration in Earth's atmosphere in the year 2035 according to the quadratic model?
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Based on the linear model for 1980–2007, was the 1990 prediction for the year 2035 accurate?
Based on the linear model for 1980–2007, was the 1990 prediction for the year 2035 accurate?
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What is the interpretation of the slope of a line when the x- and y-axes have the same unit of measure?
What is the interpretation of the slope of a line when the x- and y-axes have the same unit of measure?
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What is stated about two distinct nonvertical lines being parallel?
What is stated about two distinct nonvertical lines being parallel?
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What is the interpretation of the slope of a line when the x- and y-axes have different units of measure?
What is the interpretation of the slope of a line when the x- and y-axes have different units of measure?
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What is stated about two nonvertical lines being perpendicular?
What is stated about two nonvertical lines being perpendicular?
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Lines parallel to the line 2x - 3y = 5 have a slope of ___.
Lines parallel to the line 2x - 3y = 5 have a slope of ___.
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The slope of a line is the ratio of its height (the rise) to the length of its base (the _____).
The slope of a line is the ratio of its height (the rise) to the length of its base (the _____).
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How can you determine if two nonvertical lines are parallel?
How can you determine if two nonvertical lines are parallel?
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What is the slope of a line perpendicular to the line 2x - 3y = 5?
What is the slope of a line perpendicular to the line 2x - 3y = 5?
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Find the equation of the vertical line with an x-intercept at 3.
Find the equation of the vertical line with an x-intercept at 3.
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Show that the line with intercepts (a, 0) and (0, b) has the equation x/a + y/b = 1 (a ≠ 0, b ≠ 0).
Show that the line with intercepts (a, 0) and (0, b) has the equation x/a + y/b = 1 (a ≠ 0, b ≠ 0).
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When is the line represented by ax - by = 4 parallel to the x-axis?
When is the line represented by ax - by = 4 parallel to the x-axis?
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When is the line represented by ax - by = 4 parallel to the y-axis?
When is the line represented by ax - by = 4 parallel to the y-axis?
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Study Notes
Preparation for Calculus
- Reviewing concepts helpful for calculus study
- Sketching graphs of equations and functions
- Fitting mathematical models to data
- Key concepts to learn:
- Identifying characteristics of equations and sketching their graphs
- Finding and graphing equations of lines, including parallel and perpendicular lines
- Evaluating and graphing functions and their transformations
- Fitting mathematical models to real-life data sets
Graphs and Models
- The graph of an equation is the set of all solution points
- Point plotting: a method to sketch a graph by plotting points and connecting them with a smooth curve
- Analytic approach: solving an equation for y to find the graph
- Graphical approach: using a graphing utility to graph an equation
- Importance of choosing a suitable viewing window when using a graphing utility
Intercepts of a Graph
- Intercepts: points at which the graph intersects the x- or y-axis
- x-intercepts: points of the form (a, 0)
- y-intercepts: points of the form (0, b)
- Methods to find intercepts:
- Analytic approach: letting y = 0 to find x-intercepts and x = 0 to find y-intercepts
- Graphical approach: using a graphing utility to approximate intercepts
Symmetry of a Graph
- Three types of symmetry:
- Symmetry with respect to the y-axis
- Symmetry with respect to the x-axis
- Symmetry with respect to the origin
- Tests for symmetry:
- Replacing x by -x yields an equivalent equation for y-axis symmetry
- Replacing y by -y yields an equivalent equation for x-axis symmetry
- Replacing x by -x and y by -y yields an equivalent equation for origin symmetry
- Importance of knowing symmetry when sketching graphs### Symmetry of Polynomial Graphs
- A polynomial graph has symmetry with respect to the y-axis if each term has an even exponent (or is a constant).
- A polynomial graph has symmetry with respect to the origin if each term has an odd exponent.
Testing for Symmetry
- To test for symmetry with respect to the y-axis, replace x with -x and see if the equation remains equivalent.
- To test for symmetry with respect to the origin, replace x with -x and y with -y and see if the equation remains equivalent.
Examples of Symmetry
- The graph of y = 2x^4 + x^2 + 2 has symmetry with respect to the y-axis.
- The graph of y = 2x^3 - x has symmetry with respect to the origin.
Points of Intersection
- A point of intersection of two graphs is a point that satisfies both equations.
- To find the point(s) of intersection, solve the equations simultaneously.
Example of Finding Points of Intersection
- Find the points of intersection of the graphs of x^2 - y = 3 and x - y = 1.
- Solve the first equation for y: y = x^2 - 3.
- Solve the second equation for y: y = x - 1.
- Equate the two expressions for y and solve for x.
- Substitute the values of x into either of the original equations to find the corresponding values of y.
Mathematical Models
- Mathematical models are used to represent real-life data.
- A good model should strive for two conflicting goals: accuracy and simplicity.
- Examples of mathematical models include quadratic and linear models.
Comparing Two Mathematical Models
- A quadratic model and a linear model can be used to predict the carbon dioxide level in the atmosphere.
- The quadratic model was used to predict the level in 2035, but the linear model suggests that the prediction was too high.
Exercises
- Exercises 1-4 involve matching equations with their graphs.
- Exercises 19-28 involve finding intercepts of various equations.
- Exercises 29-40 involve testing for symmetry with respect to each axis and the origin.
- Exercises 41-58 involve sketching the graph of an equation and identifying intercepts and symmetry.Here are the study notes on the text:
Slope of a Line
- The slope of a nonvertical line is a measure of the number of units the line rises (or falls) vertically for each unit of horizontal change from left to right.
- The slope of a nonvertical line is calculated using the formula: m = (y2 - y1) / (x2 - x1)
Properties of Slope
- If the slope is positive, the line rises from left to right.
- If the slope is zero, the line is horizontal.
- If the slope is negative, the line falls from left to right.
- If the slope is undefined, the line is vertical.
Equations of Lines
- Any two points on a nonvertical line can be used to calculate its slope.
- The point-slope equation of a line is given by: y - y1 = m(x - x1)
- This equation can be rewritten in the form y = y1 + m(x - x1)
Interpreting Slope
- The slope of a line can be interpreted as either a ratio or a rate.
- Slope represents the change in y (vertical change) over the change in x (horizontal change).
Example: Finding an Equation of a Line
- Find an equation of the line that has a slope of 3 and passes through the point (1, -2).
- Use the point-slope form: y - (-2) = 3(x - 1)
- Simplify: y = 3x - 5
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Description
Review key concepts before studying calculus, including graphing equations and functions, and fitting mathematical models to data.
