Linear Functions

Big Idea Set-builder notation is more versatile than roster notation. In roster notation you must be able to list ALL elements or unambiguously imply them (with …). In set-builder notation you only need to specify the rule that the elements must specify. Goals Translate between roster and set-builder notation in the description of a set.

Linear Functions Big Idea Relationships between elements of two sets can be one-to-one (like a student ID). Relationships between the elements of two sets can be many-to-one (like gender or age). Goals Describe a mathematical relationship as a set of ordered pairs from the domain and range. many-to-one one-to-one

Linear Functions Big Idea The relationship between elements in a set can be described with a set of ordered pairs. In an ordered pair (a,b) the first of the pair (a) represents an element in one set and the second of the pair (b) represents an element in the second set. Goals Describe a mathematical relationship as a set of ordered pairs from the domain and range.

Linear Functions Goals Describe a mathematical relationship as a set of ordered pairs from the domain and range. Even Bigger Idea The set of all possible first values among the ordered pairs of a relationship is called the domain. The set of all possible second values is called the range. Here the domain is the set of all students that have been given an ID. The range is the set of all IDs.

Linear Functions Goals Define a function in terms of a restriction on the number of ordered pairs having the same elements of the domain. And The Biggest Idea A function is a relationship in which no two ordered pairs have the same first value. You can see how important it is to have just one name attached to just one ID. If two ordered pairs existed in which the first value was the same then that person would have two IDs.

Linear functions HW Due 1/ , 345.2, 13, 15, 16, 19, 27, 29,31, 33, 34, 37, , 41 A rate is an example of a function. The cost of one can of soup is $1.39. So the cost of n cans is $1.39 x n. Let C be the cost. Then in set-builder notation the set of ordered pairs is {(n,C)|C=$1.39*n} And in roster notation {(1,$1.39), (2, $2.78), (3, $3.87), …}

Linear functions HW Due 1/ , 345.2, 13, 15, 16, 19, 27, 29, 31, 33, 34, 37, , 41 You can test to see if an ordered pair satisfies a relationship: In y=5x, is the ordered pair (3,15) a solution? In y=3x+1, is the ordered pair (7,22) a solution? In 3x-2y=0, is (3,2) a solution? In 3y>2x+1, is (4,3) a solution? In 5x-2y<19, is (3,-2) a solution?

Linear functions HW Due 1/ , 345.2, 13, 15, 16, 19, 27, 29, 31, 33, 34, 37, , 41 In a function there can only be one ordered pair that has a particular first value: Is {(1,2),(2,1),(3,4),(4,3)} a function? Is {(81,9),(81,-9),(25,5),(2-5,3)} a function?

Linear functions HW Due 1/ , 345.2, 13, 15, 16, 19, 27, 29, 31, 33, 34, 37, , 41 The range is defined by the domain and the function: a. If {x|10<x<13 and x is an integer} what is the range of the relationship y=x/6? b. If {6x|x is a whole number} what is the range of the relationship y=x/6? (in roster notation the domain is {6,12,18,24, …}

Linear functions HW Due 1/ , 345.2, 13, 15, 16, 19, 27, 29, 31, 33, 34, 37, , 41 The range is defined by the domain and the function: a. If {x|10<x<13 and x is an integer} what is the range of the relationship y=-x-2? b. If {6x|x is a whole number} what is the range of the relationship y=-x-2? (in roster notation the domain is {6,12,18,24, …} That leaves three problems for you!

Linear functions HW Due 1/ , 345.2, 13, 15, 16, 19, 27, 29, 31, 33, 34, 37, , In an isosceles triangle whose perimeter is 54 centimeters, the length of the base in cm is x and the length of each leg in cm is y. Some ordered pairs in the solution set

Linear functions HW Due 1/ , 345.2, 13, 15, 16, 19, 27, 29, 31, 33, 34, 37, , At the beginning of the day, the water in the pool was 2 feet deep. Water is added so that the depth increases by less than 0.4 foot per hour. What was the depth, y, after being filled for x hours? Some ordered pairs in the solution set

Linear functions An amusement park, rides are paid for with tokens. Some rides require two tokens, and other require one. If x is the number of 2-token rides and y is the number of 1-token rides, what rides can be taken with 10 tokens? This is also the relationship when 10 feet of fence are used to enclose three sides of a rectangle pen attached to a garage where x is the number of feet perpendicular to the garage and y is the number of feet parallel to the garage. All solutions: Some pairs:

Linear functions Big Idea A linear equation in standard form is written Ax+By=C where A, B, and C are real numbers and A and B are not both zero. Goal Express a linear function symbolically in standard form and construct and interpret the graph. The preceding example of a equality describing the relationship between the sides and base of an isosceles triangle whose perimeter is 54 cm, is a linear relationship in the standard form.; 2y+x=54

Linear functions Goal Express a linear function symbolically in standard form and construct and interpret the graph. Big Idea To make a graph of an equation in standard form: solve Ax+By=C for y: y=(C/B)-(A/B)x select three values of x from the domain evaluate the equation for y to get three ordered pairs in the coordinate plane, locate these pairs draw a straight line that passes through the three points

Goal Express a linear function symbolically in standard form and construct and interpret the graph. Big Idea To make a graph of an equation in standard form: solve Ax+By=C for y: y=(C/B)-(A/B)x select three values of x from the domain evaluate the equation for y to get three ordered pairs in the coordinate plane, locate these pairs draw a straight line that passes through the three points

Goal Express a linear function symbolically in standard form and construct and interpret the graph.

Linear functions Goal Express a linear function symbolically in standard form and construct and interpret the graph. Big Idea To make a graph of an equation in standard form: solve Ax+By=C for y: y=(C/B)-(A/B)x select three values of x from the domain evaluate the equation for y to get three ordered pairs in the coordinate plane, locate these pairs draw a straight line that passes through the three points Graph these on a single graph with the domain [-2.5,10]

HW Due 1/ , 3, 5, 6, 8, 13, , 25, 28, 29, 35, 36, 46, 48, 52, 53, 55, , 14, 15, 16, 18e, c, 19e, 24

Linear functions Goal Express a linear function symbolically in standard form and construct and interpret the graph. Big Idea To make a graph of an equation in standard form: solve Ax+By=C for y: y=(C/B)-(A/B)x select three values of x from the domain evaluate the equation for y to get three ordered pairs in the coordinate plane, locate these pairs draw a straight line that passes through the three points Graph these on a single graph with the domain [-1,1]

Linear functions Goal Express a linear function symbolically in standard form and construct and interpret the graph. Big Idea To make a graph of an equation in standard form: solve Ax+By=C for y: y=(C/B)-(A/B)x select three values of x from the domain evaluate the equation for y to get three ordered pairs in the coordinate plane, locate these pairs draw a straight line that passes through the three points Graph these on a single graph with the domain [-1,1]

Linear functions Goal Express a linear function symbolically in standard form and construct and interpret the graph. Big Idea Standard form: Ax+By=C Two cases of the standard form are especially important. These are when A=0 and the graph is a horizontal line. The other is when B=0 and the graph is a vertical line. Graph these on a single graph with where 0≤x≤3 and 0≤y≤3

Big Idea Standard form: Ax+By=C Two cases of the standard form are especially important. These are when A=0 and the graph is a horizontal line. The other is when B=0 and the graph is a vertical line.

Linear functions Goal Calculate the slope of a linear relationship using ordered pairs. Big Idea The slope of a linear relationship is the ratio of the change in the y-values and the change in the x-values. A pair of ordered pairs can be used to calculate the slope. For two ordered pairs Example: Given (1,3) and (2,4), the slope =(4-3)/(2-1)=1

Linear functions Goal Calculate the slope of a linear relationship using ordered pairs. Big Idea In a linear relationship if the dependent variable, y, increases as the independent variable, x, increases, then the slope is positive.

Linear functions Goal Calculate the slope of a linear relationship using ordered pairs. Big Idea In a linear relationship if the dependent variable, y, doesn’t change as the independent variable, x, increases then the slope is 0; the graph is parallel to the x-axis. In a linear relation if the independent variable, x, doesn’t change as the dependent variable, y, increases then the slope is undefined; the graph is parallel to the y-axis. HW Due 1/ , , 362.9, 13, 14, 19, 26, 29, 30, 31, , 35

Linear functions Goal Calculate the slope of a linear relationship using ordered pairs. HW Due 1/ , , 362.9, 13, 14, 19, 26, 29, 30, 31, , Draw a line with a slope of 2 that goes through the ordered pair (0,0).

Linear functions Goal Calculate the slope of a linear relationship using ordered pairs. HW Due 1/ , , 362.9, 13, 14, 19, 26, 29, 30, 31, , Draw a line with a slope of -1/3 that goes through the ordered pair (-2,3).

Linear functions Goal Calculate the slope of a linear relationship using ordered pairs. HW Due 1/ , , 362.9, 13, 14, 19, 26, 29, 30, 31, , Plot the points (3,- 2), (9,-2), (7,4), and (1,4). What is the shape? What are the slopes of the sides?

Linear functions Big Idea If two lines are parallel they have the same slope. If two lines have the same slope they are parallel. If two lines are perpendicular their slopes are negative reciprocals. If the slopes of two lines are negative reciprocals they are are perpendicular. If a = -1/b then b is the negative reciprocal of a and ab=-1. HW Due 1/ , 3, 5, 12,

Linear functions Goal Express a linear function symbolically in intercept form and construct and interpret the graph. Big Idea If the slope of a linear equation is neither undefined (vertical line) or 0 (horizontal line) then the graph of the equation crosses the y-axis at the y-intercept (where x=0) and crosses the x-axis at the x-intercept (where y=0).

Linear functions Goal Express a linear function symbolically in intercept form and construct and interpret the graph. Big Idea The intercept form of a linear equation is obtained by dividing the standard form of a linear equation, Ax+By=C, by C. The x and y-intercepts are seen in the ratios formed with x and y. x-intercept y-intercept

Linear functions Goal Express a linear function symbolically in intercept form and construct and interpret the graph. x-intercept y-intercept Dividing by a fraction, C/A, is the same thing as multiplying by the inverse of the fraction. The inverse of C/A is A/C. So

Linear functions Goal Express a linear function symbolically in intercept form and construct and interpret the graph. Example In standard form this linear equation is transformed to intercept form: x-intercept y-intercept

Linear functions Goal Express a linear function symbolically in slope-intercept form and construct and interpret the graph. Big Idea The slope-intercept form of a linear equation is obtained by solving the standard form of a linear equation, Ax+By=C, for y. In the slope-intercept form the coefficient of x is the slope and the ratio C/B is the y-intercept. slope y-intercept

Linear functions Goal Translate among standard, intercept, and slope-intercept forms of a linear equation. Express the linear equation in standard form in intercept and slope-intercept forms: Standard form Intercept form Slope-intercept form

Linear functions Goal Translate among standard, intercept, and slope-intercept forms of a linear equation. Express the linear equation in standard form in intercept and slope-intercept forms: Standard form Intercept form Slope-intercept form

Linear functions Goal Translate among standard, intercept, and slope-intercept forms of a linear equation. Express the linear equation in standard form in intercept and slope-intercept forms: Standard form Intercept form Slope-intercept form

Linear functions Goal Translate among standard, intercept, and slope-intercept forms of a linear equation. Express the linear equation in slope-intercept form in standard and intercept forms: Standard form Intercept form Slope-intercept form

Linear functions Goal Translate among standard, intercept, and slope-intercept forms of a linear equation. Express the linear equation in standard form in intercept and slope-intercept forms: Standard form Intercept form Slope-intercept form

Linear functions Goal Translate among standard, intercept, and slope-intercept forms of a linear equation. Express the linear equation in slope-intercept form in standard and intercept forms: Standard form Intercept form Slope-intercept form

Linear functions Goal Translate among standard, intercept, and slope-intercept forms of a linear equation. Express the linear equation in standard form in intercept and slope-intercept forms: Standard form Intercept form Slope-intercept form

Linear functions Making a quick plot of a slope-intercept form plot the y-intercept draw a line through the y-intercept with the slope

Linear functions Making a quick plot of an intercept form plot the y-intercept plot the x-intercept draw a line to connect them HW Due 1/ , 2, 9, 14, 17, 18, 20, 23, 24

Linear functions Finding the linear equation through a pair of points: HW Due 1/ , 2, 9, 14, 17, 18, 20, 23, What is the linear equation with solutions (-2,3) and (1,-3)? Is the ordered pair (0,-1) a solution? calculate the slope use the slope-intercept form to find the intercept

Linear functions Goal Describe the effect of translation, reflection, and scaling on the graph of a linear function. Translation: When a value is added to right hand side of a linear equation the line is shifted up (positive value added) or down (negative value added).

Linear functions Goal Describe the effect of translation, reflection, and scaling on the graph of a linear function. Reflection: When the x value is replace by its opposite the line is reflected.

Linear functions Goal Describe the effect of translation, reflection, and scaling on the graph of a linear function. Reflection: When the x value is replaced by its opposite the line is reflected over the y-axis.

Linear functions Goal Describe the effect of translation, reflection, and scaling on the graph of a linear function. Scaling: When the x value is replaced by a constant multiplied by x the graph is contracted (the constant is less than 1 and greater than 0) or expanded (the constant is greater than 1). Replacing x by x/2 is equivalent to replacing m by m/2.

Linear functions Goal Describe the effect of translation, reflection, and scaling on the graph of a linear function. HW Due 1/ , 11, 14, 21, 22, 23, 24, 25, 26, 27 Describe how y=x has been translated, reflected and/or scaled and sketch a graph

Linear functions Goal Describe the effect of translation, reflection, and scaling on the graph of a linear function. HW Due 1/ , 11, 14, 21, 22, 23, 24, 25, 26, 27 Indicate if y=x has been translated, reflected and/or scaled and sketch a graph translate translate & reflect scale

Linear functions Goal Describe direction variation as a slope-intercept form of a linear equation in which the y-intercept is 0. Big Idea y=mx is a direct variation. The slope of the line is the constant of variation. HW Due 1/ , 377.6, 12, 13, 15, 16, Example A printer can print 160 characters (y, the dependent variable) in 10 seconds (x, the independent variable).

Linear functions Goal Solve narrative problems by applying the concept of the slope.