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Chapter 1

Chapter 1 . Linear Functions. Slopes and Equations of Lines. The Rectangular Coordinate System The horizontal number line is the x-axis The vertical number line is the y-axis The point where the axes intersect is the origin (the 0 point). Slopes and Equations of Lines.

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Chapter 1

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  1. Chapter 1 Linear Functions

  2. Slopes and Equations of Lines • The Rectangular Coordinate System • The horizontal number line is the x-axis • The vertical number line is the y-axis • The point where the axes intersect is the origin (the 0 point)

  3. Slopes and Equations of Lines • The point given as (2,3) is also called an ordered pair. • Where 2 is the x coordinate, the first coordinate, the abscissa, or the horizontal distance • Where 3 is the y coordinate, the second coordinate, the ordinate, or the vertical distance

  4. Slopes and Equations of Lines • Intercepts of a Line • The y-intercept of a line is the point (0,b) where the line intersects the y axis. To find b, substitute 0 for x in the equation of the line and solve for y. • The x-intercept of a line is the point (a,0) where the line intersects the x-axis. To find a, substitute 0 for y in the equation of a line and solve for x.

  5. Slopes and Equations of Lines • Slope • Slope (m) is a measure of the slant of a line. • The slope of the line between two points is the ratio of the change in y to the change in x.

  6. Slopes and Equations of Lines • The slope of a line is defined as the vertical change (the “rise”) over the horizontal change (the “run”) as one travels along the line.

  7. Slopes and Equations of Lines • Slope of a Line

  8. Slopes and Equations of Lines • Point-Slope Form of the Equation of a Line • Recall the slope-intercept from of a line is • Y = mx + b • where m is slope and b is the y-intercept CASE A GIVEN A POINT AND THE SLOPE • The equation of the line passing through P(x1,y1) and with slope m is • y – y1 = m( x – x1 ) (the point-slope formula) • So given a slope and a point (other than y-intercept) • We replace m with the given slope • We replace x1 and y1 with the point coordinates • Simplify • Write the equation in the form of y = mx + b

  9. Slopes and Equations of Lines CASE B GIVEN TWO POINTS ONLY • The equation of the line passing through P(x1,y1) and P(x2,y2) (slope is not given) • We first use the slope formula to determine slope • Then we use one of the given points and the slope in the point-slope formula y – y1 = m( x – x1 )

  10. Slopes and Equations of Lines CASE C GIVEN A POINT P(x1,y1) AND AN UNDEFINED SLOPE • Recall that a line with an undefined slope is a vertical line with x = a, where a is the x-coordinate of the x-intercept (x1, 0) • Recall that for a vertical line, the x-coordinate doesn’t change. •  the equation would simply be x = x1

  11. Slopes and Equations of Lines CASE D GIVEN A POINT P(x1,y1) AND A ZERO SLOPE • Recall that a line with a zero slope is a horizontal line with y = b, where b is the y-coordinate of the y-intercept (0, b) • Recall that for a horizontal line, the y-coordinate doesn’t change. •  the equation would simply be y = y1

  12. Slopes and Equations of Lines CASE E GIVEN THE Y-INTERCEPT (0,b) AND SLOPE • We use the slope-intercept form • y = mx +b • Replace m with the given slope • Replace b with the given y-coordinate

  13. Slopes and Equations of Lines • Slopes of Horizontal and Vertical Lines • horizontal lines (y = b) have a slope of 0 • vertical lines (x = a) have no defined slope • Slopes of Parallel Lines • Two nonvertical lines are parallel if they have the same slope (they do not intersect) • (i.e., line A has slope of 1/2 and line B has a slope of 1/2) • Slopes of Perpendicular Lines • Two nonvertical lines are perpendicular if their slopes are negative reciprocals of each other. These lines intersect at right angles • (i.e., line A has slope of 1/2 and line B has a slope of -2)

  14. Slopes and Equations of Lines • Finding equations of parallel and perpendicular lines • To find equations of parallel and perpendicular lines we always need to know slope. • Recall that • parallel lines () – two lines that do not intersect - have the same slope. • perpendicular lines (  ) – two lines that intersect at right angles – have slopes that are negative reciprocals of each other.

  15. Slopes and Equations of Lines • Example • Find the equation of a line containing the point (3,2) and  to the line 3x + y = -3. • To be parallel, the lines must have the same slope. The line 3x + y = -3 has a slope of -3 because when we solve for y : y = -3x -3 • Given a point (3,2) and a slope -3 we use the point-slope formula • y – y1 = m( x – x1 ) • The equation is y = -3x + 11

  16. Slopes and Equations of Lines • Example • Find the equation of a line containing the point (-1,-3) and to the line 3x - 5y = 2. • To be perpendicular, the lines must have slopes that are negative reciprocals of each other. The line 3x - 5y = 2 has a slope of 3/5 because when we solve for y : • y = 3/5x -2/5 • The slope of the other line has to be -5/3 • Given a point (-1,-3) and a slope -5/3 we use the point-slope formula • y – y1 = m( x – x1 ) • The equation is y = -5/3x – 14/3

  17. Slopes and Equations of Lines • Graphing Lines • To graph an equation in two variables such as y = x – 1 when x = -2, -1, 0, 1, 2. • Using Substitution • Substitute each value of x into the equation. • Solve for y • Graph the ordered pairs • Using slope and y-intercept • Place a point at the y-intercept • Starting from the y-intercept, “rise” then “run” based on slope

  18. Linear Functions And Applications • Linear Functions • Notation – f(x) • Note: f, g, or h are often used to name functions • The function f(x) is defined by • y = f(x) = mx + b

  19. Linear Functions And Applications • Operations • The sum of f and g, denoted as f + g, is defined by (f + g)(x) = f(x) + g(x) • The difference of f and g, denoted as f – g, is defined by (f – g)(x) = f(x) – g(x) • The product of f and g, denoted as f∙g, is defined by (f∙g)(x) = f(x)g(x) • The quotient of f and g, denoted as f/g, is defined by (f/g)(x) = where g(x)0 • Linear Functions

  20. Linear Functions And Applications • Example • Let f(x) = 4x – 1. Find f(-2), f(-1), f(0), f(1), f(2) f(-2) = 4(-2) – 1 = -8 – 1 = -9 f(-1) = 4(-1) – 1 - -4 – 1 = -5 f(0) = 4(0) – 1 = 0 -1 = -1 f(1) = 4(1) – 1 = 4 – 1 = 3 f(2) = 4(2) – 1 = 8 – 1 = 7

  21. Linear Functions And Applications • Supply and Demand • Supply and Demand functions are not necessarily linear • Most approximately linear • Equilibrium Price/Quantity • Equilibrium price of a commodity is the price found at the point where the supply and demand are equal (i.e., where the supply and demand graphs for that commodity intersect). • Equilibrium quantity is the quantity where the prices from both demand and the supply are equal.

  22. Linear Functions And Applications • Cost Analysis • The cost of manufacturing an item commonly consists of two parts: • Fixed cost – designing the product, setting up a factory, training workers, etc. • Marginal cost – approximates the cost of producing one additional item.

  23. Linear Functions And Applications • Cost Analysis • The Cost Function • C(x) = mx + b where m represents marginal cost per item where b represents the fixed cost

  24. Linear Functions And Applications • Break-Even Analysis • Involves both revenue and profit. • Revenue R(x) = px where p represents the price per unit of product where x represents the number of units sold (demand) • Profit P(x) is the difference between revenue and cost: • P(x) = R(x) – C(x) • Break-even quantity – the number of units at which revenue just equals cost • Break-even point – the point where revenue equals cost.

  25. The Least Squares Line • In practice, how are equations of supply and demand functions found? • Data is collected • Data is plotted • Data lie perfectly along a line – equation easily found using any two points • Data scattered and no line that goes through all points – must find a line that approximates the linear trend of the data as closely as possible (Method of Least Squares)

  26. The Least Squares Line • How do we know when we have the best line (least squares line) ? (see Figure 16, p. 30) • The line whose sum of squared vertical distances from the data points is as small as possible. • The least squares line Y = mx + b, where and or

  27. The Least Squares Line • Calculating by hand • Given “n” pairs of data, find the required sums. x y xy x2 y2 _________________________________________________ xy xy x2y2 • Plug and Chug • Write least squares equation in the form of Y = mx + b.

  28. The Least Squares Line • Coefficient of Correlation • Measures of “goodness of fit” of the least squares line

  29. The Least Squares Line • The coefficient of correlation (r) is always equal to or between 1 and -1. • Coefficient of correlation (r) values of exactly 1 or -1 mean that the data points lie exactly on the least squares line. • If r = 1, the least squares line has a positive slope • If r = -1, the least squares line has a negative slope • If r = 0, there is no linear correlation between the data points (some nonlinear function might provide a better fit) • The closer the value of r is to 1 or -1, the stronger the linear relationship between the data points.

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