Graphs, Linear Equations, and Functions

Transcription

1 Graphs, Linear Equations, and Functions. The Rectangular R. Coordinate Fractions Sstem bjectives. Interpret a line graph.. Plot ordered pairs.. Find ordered pairs that satisf a given equation. 4. Graph lines.. Find - and -intercepts.. Recognize equations of horizontal and vertical lines and lines passing through the origin. 7. Use the midpoint formula. Slide Section of 04., Slide Section of 04., Rectangular (or Cartesian, for Descartes) Coordinate Sstem Rectangular (or Cartesian, for Descartes) Coordinate Sstem rigin Quadrant II 4 Quadrant I Quadrant III 8-4 Quadrant IV - -ais -ais D rdered Pair (, ) A A (, ) B (, ) B C C (, ) D ( 4, ) Quadrant Quadrant I Quadrant IV Quadrant III Quadrant II -8 Slide Section of 04., Slide 4 Section of 04., 4 Caution EXAMPLE Completing rdered Pairs CAUTIN The parentheses used to represent an ordered pair are also used to represent an open interval (introduced in Section.). The contet of the discussion tells whether ordered pairs or open intervals are being represented. Complete each ordered pair for + 4 = 7. (a) (,? ) We are given =. We substitute into the equation to find. + 4 = 7 () + 4 = 7 Let =. + 4 = 7 4 = 8 = The ordered pair is (, ). Slide Section of 04., Slide Section of 04.,

2 Completing rdered Pairs Complete each ordered pair for + 4 = 7. (b)(?, ) Replace with in the equation to find. + 4 = 7 + 4( ) = 7 Let =. 0 = 7 = 7 = 9 The ordered pair is (9, ). A Linear Equation in Two Variables A linear equation in two variables can be written in the form A + B = C, where A, B, and C are real numbers (A and B not both 0). This form is called standard form. Slide 7 Section of 04., 7 Slide 8 Section of 04., 8 Intercepts Finding Intercepts -intercept (where the line intersects the -ais) -intercept (where the line intersects the -ais) When graphing the equation of a line, find the intercepts as follows. let = 0 to find the -intercept; let = 0 to find the -intercept. Slide 9 Section of 04., 9 Slide 0 Section of., 04 EXAMPLE Finding Intercepts Find the - and -intercepts of =, and graph the equation. We find the -intercept b letting = 0. = 0 = Let = 0. = = -intercept is (, 0). We find the -intercept b letting = 0. = (0) = Let = 0. = = -intercept is (0, ). Finding Intercepts Find the - and -intercepts of =, and graph the equation. The intercepts are the two points (,0) and (0, ). We show these ordered pairs in the table net to the figure below and use these points to draw the graph. 0 0 The intercepts are the two points (,0) and (0, ). Slide Section of., 04 Slide Section of., 04

3 EXAMPLE Graphing a Horizontal Line Graphing a Vertical Line Graph =. Since is alwas, there is no value of corresponding to = 0, so the graph has no -intercept. The -intercept is (0, ). The graph in the figure below, shown with a table of ordered pairs, is a horizontal line. Graph + =. The -intercept is (, 0). The standard form + 0 = shows that ever value of leads to =, so no value of makes = 0. The onl wa a straight line can have no -intercept is if it is vertical, as in the figure below. 0 0 Slide Section of., 04 Slide 4 Section of., 044 Horizontal and Vertical Lines EXAMPLE 4 Graphing a Line That Passes through the rigin CAUTIN To avoid confusing equations of horizontal and vertical lines, keep the following in mind.. An equation with onl the variable will alwas intersect the -ais and thus will be vertical. It has the form = a.. An equation with onl the variable will alwas intersect the -ais and thus will be horizontal. It has the form = b. Graph + = 0. We find the -intercept b letting = 0. + = = 0 Let = 0. = 0 = 0 -intercept is (0, 0). We find the -intercept b letting = 0. + = 0 (0) + = 0 Let = = 0 = 0 -intercept is (0, 0). Both intercepts are the same ordered pair, (0, 0). (This means the graph goes through the origin.) Slide Section of., 04 Slide Section of., 04 Graph + = 0. Graphing a Line That Passes through the rigin To find another point to graph the line, choose an nonzero number for, sa =, and solve for. Let =. + = 0 () + = 0 Let =. + = 0 = This gives the ordered pair (, ). Graph + = 0. Graphing a Line That Passes through the rigin These points, (0, 0) and (, ), lead to the graph shown below. As a check, verif that (, ) also lies on the line intercept and -intercept Slide 7 Section of., 047 Slide 8 Section of., 048

4 Use the midpoint formula If the endpoints of a line segment PQ are (, ) and (, ), its midpoint M is,. EXAMPLE Finding the Coordinates of a Midpoint Find the coordinates of the midpoint of line segment PQ with endpoints P(, ) and Q(4, ). Use the midpoint formula with =, = 4, =, = : 4 ( ), 0,, Midpoint Slide 9 Section of., 049 Slide 0 Section of., 040. The Slope of a R. Line Fractions bjectives. Find the slope of a line, given two points on the line.. Find the slope of a line, given an equation of the line.. Graph a line, given its slope and a point on the line. 4. Use slopes to determine whether two lines are parallel, perpendicular, or neither.. Solve problems involving average rate of change. Slide Section of., 04 Find the Slope of a Line Given Two Points on the Line ne of the important properties of a line is the rate at which it is increasing or decreasing. The slope is the ratio of vertical change, or rise, to horizontal change, or run. As we move from P to P : 4 ft P ft P Slide Section of., 04 Find the Slope of a Line Given Two Points on the Line Eample Finding the Slope of a Line Find the slope of the line containing the points (, ) and (, ). (,), (,), m ( ) Rise = = Run = ( ) = There is a rise of unit for a run of units. Slide Section of., 04 Slide 4 Section of., 044 4

5 Find the Slope of a Line Given the Equation of the Line Eample Find the slope of the line 4 = 8. The intercepts can be used as the two points needed to find the Let = 0 to find that the -intercept is (, 0). Let = 0 to find that the -intercept is (0, 8). m ( ) 8 or 4 Slide Section of., 04 Finding the Slope of Horizontal and Vertical Lines Eample Find the slope of each line. a. = The graph of = is a horizontal line. To find the slope, select two different points on the line, such as (, ) and (0, ) and use the slope formula. 0 m 0 The rise is 0, so the slope is 0. 0 Slide Section of., 04 Finding the Slope of Horizontal and Vertical Lines Eample Eample 4 Finding the Slope from an Equation Find the slope of each line. b. = The graph of = is a vertical line. To find the slope, select two different points on the line, such as (, ) and (, 0) and use the slope formula. m 0 Since division b 0 is undefined, the slope is undefined. 0 Slide 7 Section of., 047 Find the slope of the graph = 8. Solve the equation for. The graph of = is a vertical line The slope is the coefficient of, so the slope is /. Slide 8 Section of., 048 Eample Using the Slope and a Point to Graph a Line rientation of a Line in the Plane Graph the line with slope / and through the point (, ). Locate the point P(, ). From the slope formula: m change in change in So, move down units and then units to the right to the point R(, ). Down P Right R Slide 9 Section of., 049 Slide 0 Section of., 040

6 Slopes of Parallel and Perpendicular Lines Eample Determining Whether Two Lines are Parallel Determine whether the lines passing through (, ) and (4, ) and through (, 0) and (0, ) are parallel. Find the slope of each line. 4 m 4 ( ) 0 m 0 Because the slopes are equal, the two lines are parallel. Slide Section of., 04 Slide Section of., 04 Slopes of Parallel and Perpendicular Lines Determining Whether Two Lines are Perpendicular Eample 7 Are the lines with equations = and + = perpendicular? A line with slope 0 is perpendicular to a line with undefined slope. Find the slope of each line b solving each equation for. The slopes are negative reciprocals because their product is. The lines are perpendicular. Slide Section of., 04 Slide 4 Section of., 044 Eample 8 Determining Whether Two Lines Are Parallel, Perpendicular or Neither Decide whether each pair of lines is parallel, perpendicular, or neither. 8 = 4 and + = Find the slope of each line b first solving each equation for. 8 = 4 + = 8 8 = = Slope is. = 4 Slope is 4. Slide Section of., 04 Determining Whether Two Lines Are Parallel, Perpendicular or Neither Eample 8 (continued) Decide whether each pair of lines is parallel, perpendicular, or neither. Because the slopes are not equal, the lines are not parallel. To see if the lines are perpendicular, find the product of the slopes. 4 ( ) = 0 4 The lines are not perpendicular because the product of their slopes is not. The lines are neither parallel nor perpendicular Slide Section of., 04

7 Interpreting Slope as Average Rate of Change Eample 0 Cind purchased a new car in 00 for $8,000. In 0, the car had a value of $700. At what rate is the car s value changing with respect to time? To determine the average rate of change, we need two pairs of data. If = 00, then = 8,000 and if = 0, then = 700. average rate of change 700 8, ,00 00 This means the car decreased in value b $00 each ear from 00 to 0. Slide 7 Section of., 047. Linear Equations R. Fractions in Two Variables bjectives. Write an equation of a line, given its slope and - intercept.. Graph a line, using its slope and -intercept.. Write an equation of a line, given its slope and a point on the line. 4. Write an equation of a line, given two points on the line.. Write equations of horizontal or vertical lines.. Write an equation of a line parallel or perpendicular to a given line. 7. Write an equation of a line that models real data. Slide 8 Section of., 048 Write an equation of a line given its slope and -intercept. Write an equation of a line given its slope and -intercept. Given the slope m of a line and the -intercept b of the line, we can determine its equation. If we know the slope of a line and its -intercept, we can write its equation b substituting these values into the above equation. Slide 9 Section of., 049 Slide 40 Section of., 040 Writing an Equation of a Line Eample Find an equation of a line with slope ¾ and -intercept (0, ). m = /4 and b =. Substitute into the slope-intercept form. m b 4 Eample Graph Lines Using Slope and -Intercept Graph the line having slope / and -intercept (0, ). rise change in m run change in Plot the -intercept (0, ). Move up units and to the right units. Join the points with a straight line. Slide 4 Section of., 04 Slide 4 Section of., 04 7

8 Write an equation of a line, given its slope and a point on the line. Write an equation of a line, given its slope and a point on the line. If we know the slope m of a line and the coordinates of a point on the line, we can determine its equation. If we know the slope of a line and the coordinates of a single point on the line, we can write the equation of the line b substituting these values into the equation above. Slide 4 Section of., 04 Slide 44 Section of., 044 Finding the Equation of a Line, Given the Slope and a Point Eample Find an equation of the line having slope and passing through the point (/, ). Use the point-slope form of the equation of a line with (, ) = (/, ) and m =. m( ) Slide 4 Section of., 04 Finding an Equation of a Line, Given Two Points Eample 4 Find an equation of the line containing the points (, ) and (, ). We begin b finding the slope of the line. 4 4 m ( ) Use either point and substitute into the point-slope form of the equation of a line. 4 m( ) ( ) 4 Slide 4 Section of., 04 Writing Equations of Horizontal and Vertical Lines Equations of Horizontal and Vertical Lines Eample Write an equation of the line passing through the point (, ) that satisfies the given condition. a. The line has slope 0. Since the slope is 0, this is a horizontal line. The equation is =. b. The line has undefined slope. This is a vertical line. The equation is =. Slide 47 Section of., 047 Slide 48 Section of., 048 8

9 Writing Equations of Parallel or Perpendicular Lines Eample Write an equation in slope-intercept form of the line passing through the point (4, 7) that is parallel to the graph of + =. Find the slope of the given line b solving for. A line parallel will have the same slope. m( ) ( 7) ( 4) Slide 49 Section of., 049 Writing Equations of Parallel or Perpendicular Lines Write an equation in slope-intercept form of the line passing through the point (4, 7) that satisfies the given condition. a. The line is parallel to the graph of + =. ( 7) ( 4) ( 7) Slide 0 Section of., 040 Writing Equations of Parallel or Perpendicular Lines Eample b Write an equation in slope-intercept form of the line passing through the point (0, 0) that is perpendicular to the graph of + = 7. Find the slope of the given line b solving for A line perpendicular will have a slope of /. Writing Equations of Parallel or Perpendicular Lines Write an equation in slope-intercept form of the line passing through the point (0, 0) that is perpendicular to the graph of + = 7. Use the point (0, 0) and the point-slope form. m( ) 0 ( 0) Slide Section of., 04 Slide Section of., 04 Eample 7 Writing A Linear Equation to Describe Data A veterinarian charges $4 to visit a farm where cattle are raised. The price to vaccinate each animal is $8. Write an equation that defines the total bill that the veterinarian will submit to vaccinate all the cattle at the farm. Let denote the number of cattle to be vaccinated..4 Linear Inequalities R. Fractions in Two Variables bjectives. Graph linear inequalities in two variables.. Graph the intersection of two linear inequalities.. Graph the union of two linear inequalities. The cost of onl the vaccinations can be found b the linear equation = 8. There is a vet charge of $4 to visit the farm. The total bill can be described b = Slide Section of., 04 Slide 4 Section of., 044 9

10 Graph Linear Inequalities in Two Variables In Section., we graphed linear inequalities in one variable on the number line. In this section we learn to graph linear inequalities in two variables on a rectangular coordinate sstem. Graph Linear Inequalities in Two Variables. Step Draw the graph of the straight line that is the boundar. Make the line solid if the inequalit involves, or. Make the line dashed if the inequalit involves < or >. Step Choose a test point. Choose an point not on the line, and substitute the coordinates of this point in the inequalit. Step Shade the appropriate region. Shade the region that includes the test point if it satisfies the original inequalit. therwise, shade the region on the other side of the boundar line. Slide Section of., 04 Slide Section of., 04 Eample Graphing a Linear Inequalit Graph the inequalit +. The inequalit + means that + < or + =. The graph of + = is a line. This boundar line divides the plane into two regions. The graph of the solutions of the inequalit + < will include onl one of these regions. We find the required region b checking a test point. We choose an point not on the boundar line. Because (0, 0) is eas to substitute, we often use it. Check (0, 0) + (0) + (0) Graph Linear Inequalities Graph the inequalit +. 0 True. Since the last statement is true, we shade the region that includes the test point (0, 0). Slide 7 Section of., 047 Slide 8 Section of., 048 Graph a Linear Inequalit with Boundar Through the rigin Eample Graph the inequalit < 0. 0 Begin b graphing =, using a dashed line. Since (0, 0) is on the boundar line, choose a different test point. Here, we choose (,). < () < True Thus, we shade the region containing (,) Slide 9 Section of., 049 Eample Graph Graphing the Intersection of Two Inequalities 4 and. Graph each of the two inequalities separatel. Shade the common area. Slide 0 Section of., 040 0

11 Eample 4 Graph Graphing the Union of Two Inequalities 4 or. Graph each of the two inequalities separatel. The graph of the union includes all points in either inequalit. Shade the common area.. Introduction to R. Relations Fractions and Functions bjectives. Distinguish between independent and dependent variables.. Define and identif relations and functions.. Find domain and range. 4. Identif functions defined b graphs and equations. Slide Section of., 04 Slide Section of., 04 Independent and Dependent Variables We often describe one quantit in terms of another. We can indicate the relationship between these quantities b writing ordered pairs in which the first number is used to arrive at the second number. Here are some eamples. (, $) gallons of gasoline will cost $. The total cost depends on the number of gallons purchased. (8, $7.0) 8 gallons of gasoline will cost $7.0. Again, the total cost depends on the number of gallons purchased. Independent and Dependent Variables We often describe one quantit in terms of another. We can indicate the relationship between these quantities b writing ordered pairs in which the first number is used to arrive at the second number. Here are some eamples. (the number of gallons, the total cost) depends on Slide Section of., 04 Slide 4 Section of., 044 Independent and Dependent Variables We often describe one quantit in terms of another. We can indicate the relationship between these quantities b writing ordered pairs in which the first number is used to arrive at the second number. Here are some eamples. (0, $0) Working for 0 hours, Working for hours, (, $) ou will earn $0. The total gross pa depends on the number of hours worked. ou will earn $. The total gross pa depends on the number of hours worked. Independent and Dependent Variables We often describe one quantit in terms of another. We can indicate the relationship between these quantities b writing ordered pairs in which the first number is used to arrive at the second number. Here are some eamples. (the number of hours worked, the total gross pa) depends on Slide Section of., 04 Slide Section of., 04

12 Independent and Dependent Variables We often describe one quantit in terms of another. We can indicate the relationship between these quantities b writing ordered pairs in which the first number is used to arrive at the second number. Here are some eamples. Generalizing, if the value of the variable depends on the value of the variable, then is called the dependent variable and is the independent variable. Independent variable (, ) Dependent variable Relation Define and identif relations and functions. A relation is an set of ordered pairs. A special kind of relation, called a function, is ver important in mathematics and its applications. Function A function is a relation in which, for each value of the first component of the ordered pairs, there is eactl one value of the second component. Slide 7 Section of., 047 Slide 8 Section of., 048 Determining Whether Relations Are Functions Eample Tell whether each relation defines a function. Mapping Relations L = { (, ), (, 8), (4, 0) } M = { (, 0), (, 4), (, 7), (, 7) } N = { (, ), ( 4, 4), (, ) } Relations L and M are functions, because for each different -value there is eactl one -value. In relation N, the first and third ordered pairs have the same -value paired with two different -values ( is paired with both and ), so N is a relation but not a function. In a function, no two ordered pairs can have the same first component and different second components. Slide 9 Section of., 049 F 4 F is a function. G 0 G is not a function. Slide 70 Section of., 0470 Tables and Graphs Using an Equation to Define a Relation or Function Relations and functions can also be described using rules. Usuall, the rule is given as an equation. For eample, from the previous slide, the chart and graph could be described using the following equation. 0 0 Dependent variable = Independent variable Table of the function, F An equation is the most efficient wa to define a relation or function. Graph of the function, F Slide 7 Section of., 047 Slide 7 Section of., 047

13 NTE Functions Another wa to think of a function relationship is to think of the independent variable as an input and the dependent variable as an output. This is illustrated b the input-output (function) machine (below) for the function defined b =. Domain and Range In a relation, the set of all values of the independent variable () is the domain. The set of all values of the dependent variable () is the range. 4 (Input ) (Input ) 4 (utput ) = (utput ) Slide 7 Section of., 047 Slide 74 Section of., 0474 Eample Finding Domains and Ranges of Relations Give the domain and range of each relation. Tell whether the relation defines a function. (a) { (, 8), (, 9), (, ), (8, ) } The domain, the set of -values, is {,, 8}; the range, the set of -values, is { 8, 9,, }. This relation is not a function because the same -value is paired with two different -values, 9 and. Finding Domains and Ranges of Relations Give the domain and range of each relation. Tell whether the relation defines a function. (b) 9 The domain of this relation is {,, 9}. The range is {M, N}. This mapping defines a function each -value corresponds to eactl one -value. M N Slide 7 Section of., 047 Slide 7 Section of., 047 Finding Domains and Ranges of Relations Eample Finding Domains and Ranges from Graphs Give the domain and range of each relation. Tell whether the relation defines a function. (c) This is a table of ordered pairs, so the domain is the set of -values, {,, }, and the range is the set of -values, {}. The table defines a function because each different -value corresponds to eactl one -value (even though it is the same -value). Slide 77 Section of., 0477 Give the domain and range of each relation. (a) (, ) (0, ) (, ) (4, ) The domain is the set of -values, {, 0,, 4}. The range, the set of -values, is {,,, }. Slide 78 Section of., 0478

14 Finding Domains and Ranges from Graphs Finding Domains and Ranges from Graphs Give the domain and range of each relation. (b) Range The -values of the points on the graph include all numbers between 7 and, inclusive. The -values include all numbers between and, inclusive. Using interval notation, the domain is [ 7, ]; the range is [, ]. Give the domain and range of each relation. (c) The arrowheads indicate that the line etends indefinitel left and right, as well as up and down. Therefore, both the domain and range include all real numbers, written (-, ). Domain Slide 79 Section of., 0479 Slide 80 Section of., 0480 Finding Domains and Ranges from Graphs Give the domain and range of each relation. (d) The arrowheads indicate that the graph etends indefinitel left and right, as well as upward. The domain is (-, ). Because there is a least - value,, the range includes all numbers greater than or equal to, written [, ). Agreement on Domain The domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable. Slide 8 Section of., 048 Slide 8 Section of., 048 Vertical Line Test Eample 4 Using the Vertical Line Test If ever vertical line intersects the graph of a relation in no more than one point, then the relation represents a function. Use the vertical line test to determine whether each relation is a function. (a) (a) (b) This relation is a function. (, ) (, ) (4, ) Not a function the same -value corresponds to two different -values. Function each -value corresponds to onl one -value. (0, ) Slide 8 Section of., 048 Slide 84 Section of.,

15 Using the Vertical Line Test Using the Vertical Line Test Use the vertical line test to determine whether each relation is a function. (b) This graph fails the vertical line test since the same -value corresponds to two different -values; therefore, it is not the graph of a function. Use the vertical line test to determine whether each relation is a function. (c) This relation is a function. Slide 8 Section of., 048 Slide 8 Section of., 048 Identifing Functions from Their Equations Using the Vertical Line Test Eample Use the vertical line test to determine whether each relation is a function. (d) This relation is a function. Decide whether each relation defines a function and give the domain. (a) = In the defining equation, =, is alwas found b subtracting from. Thus, each value of corresponds to just one value of and the relation defines a function; can be an real number, so the domain is (, ). Slide 87 Section of., 0487 Slide 88 Section of., 0488 Identifing Functions from Their Equations Identifing Functions from Their Equations Decide whether each relation defines a function and give the domain. (b) = For an choice of in the domain, there is eactl one corresponding value for (the radical is a nonnegative number), so this equation defines a function. Since the equation involves a square root, the quantit under the radical sign cannot be negative. Thus, 0, and the domain of the function is [, ). Slide 89 Section of., 0489 Decide whether each relation defines a function and give the domain. (c) = The ordered pair (9, ) and (9, ) both satisf this equation. Since one value of, 9, corresponds to two values of, and, this equation does not define a function. Because is equal to the square of, the values of must alwas be nonnegative. The domain of the relation is [0, ). Slide 90 Section of., 0490

16 Identifing Functions from Their Equations Identifing Functions from Their Equations Decide whether each relation defines a function and give the domain. (d) B definition, is a function of if ever value of leads to eactl one value of. Here a particular value of, sa 4, corresponds to man values of. The ordered pairs (4, 7), (4, ), (4, ), and so on, all satisf the inequalit. Thus, an inequalit never defines a function. An number can be used for so the domain is the set of real numbers (, ). Decide whether each relation defines a function and give the domain. (e) = + 4 Given an value of in the domain, we find b adding 4, then dividing the result into. This process produces eactl one value of for each value in the domain, so this equation defines a function. The domain includes all real numbers ecept those that make the denominator 0. We find these numbers b setting the denominator equal to 0 and solving for. + 4 = 0 = 4 The domain includes all real numbers ecept 4, written (, 4) U ( 4, ). Slide 9 Section of., 049 Slide 9 Section of., 049 Variations of the Definition of Function. A function is a relation in which, for each value of the first component of the ordered pairs, there is eactl one value of the second component.. A function is a set of ordered pairs in which no first component is repeated.. A function is an equation (rule) or correspondence (mapping) that assigns eactl one range value to each distinct domain value. Slide 9 Section of., 049