Linear Functions Linear functions can be represented by tables, graphs, and equations. Being able to identify different representations of the same function is a good skill to have.

Linear Functions Linear functions can be represented by tables, graphs, and equations. Being able to identify different representations of the same function is a good skill to have. One way to connect a function to its equation is to generate some points and then graph them on the coordinate plane.

Linear Functions For example, let’s find the graph for 𝑦=−𝑥− 1

Linear Functions For example, let’s find the graph for 𝑦=−𝑥− 1 Make an x / y table using some x – values and substituting them into the rule and finding y. I like to use −2 , −1 , 0 ,1 , 2 𝒙 𝒚 −2 −1 1 2

Linear Functions For example, let’s find the graph for 𝑦=−𝑥− 1 Make an x / y table using some x – values and substituting them into the rule and finding y. I like to use −2 , −1 , 0 ,1 , 2 𝒙 𝒚 −2 1 −1 2 𝑦=− −2 −1=2−1=1

Linear Functions For example, let’s find the graph for 𝑦=−𝑥− 1 Make an x / y table using some x – values and substituting them into the rule and finding y. I like to use −2 , −1 , 0 ,1 , 2 𝒙 𝒚 −2 1 −1 2 𝑦=− −2 −1=2−1=1 𝑦=− −1 −1=1−1=0

Linear Functions For example, let’s find the graph for 𝑦=−𝑥− 1 Make an x / y table using some x – values and substituting them into the rule and finding y. I like to use −2 , −1 , 0 ,1 , 2 𝒙 𝒚 −2 1 −1 2 𝑦=− −2 −1=2−1=1 𝑦=− −1 −1=1−1=0 𝑦=− 0 −1=0−1=−1

Linear Functions For example, let’s find the graph for 𝑦=−𝑥− 1 Make an x / y table using some x – values and substituting them into the rule and finding y. I like to use −2 , −1 , 0 ,1 , 2 𝒙 𝒚 −2 1 −1 2 𝑦=− −2 −1=2−1=1 𝑦=− −1 −1=1−1=0 𝑦=− 0 −1=0−1=−1 𝑦=− 1 −1=−1−1=−2

Linear Functions For example, let’s find the graph for 𝑦=−𝑥− 1 Make an x / y table using some x – values and substituting them into the rule and finding y. I like to use −2 , −1 , 0 ,1 , 2 𝒙 𝒚 −2 1 −1 2 −3 𝑦=− −2 −1=2−1=1 𝑦=− −1 −1=1−1=0 𝑦=− 0 −1=0−1=−1 𝑦=− 1 −1=−1−1=−2 𝑦=− 2 −1=−2−1=−3

Linear Functions Then we can graph these points and create the function. 𝒙 𝒚 −2 1 −1 2 −3

Linear Functions Sometimes we will need to choose which graph represents a function. Example : Which graph below expresses the monthly expenses for a company who’s expense equation is 𝑦=4,000𝑥+4,000 32 32 ( in thousands ) Expenses ( in thousands ) Expenses 28 28 24 24 20 20 16 16 12 12 8 8 4 4 1 2 3 4 5 6 1 2 3 4 5 6 MONTH MONTH

Linear Functions Solution : 𝑦=4,000𝑥+4,000 has a y – intercept of 4,000 with a slope of 𝑚=4,000 32 32 ( in thousands ) Expenses ( in thousands ) Expenses 28 28 24 24 20 20 16 16 12 12 8 8 4 4 1 2 3 4 5 6 1 2 3 4 5 6 MONTH MONTH

Linear Functions Solution : 𝑦=4,000𝑥+4,000 has a y – intercept of 4,000 with a slope of 𝑚=4,000 Both graphs have y – intercepts of 4,000. 32 32 ( in thousands ) Expenses ( in thousands ) Expenses 28 28 24 24 20 20 16 16 12 12 8 8 4 4 1 2 3 4 5 6 1 2 3 4 5 6 MONTH MONTH

Linear Functions Solution : 𝑦=4,000𝑥+4,000 has a y – intercept of 4,000 with a slope of 𝑚=4,000 Both graphs have y – intercepts of 4,000. But only the graph on the right has a slope of 4,000. The graph on the left has a slope of 2,000. 32 32 ( in thousands ) Expenses ( in thousands ) Expenses 28 28 24 𝑚= 4000 2 =2000 24 𝑚= 4000 1 =4000 20 20 16 16 12 12 8 8 4 4 1 2 3 4 5 6 1 2 3 4 5 6 MONTH MONTH

Linear Functions Another solution method is to substitute some x – values into the given function and find y – values to create a table to check points. 32 32 ( in thousands ) Expenses ( in thousands ) Expenses 28 28 24 24 20 20 16 16 12 12 8 8 4 4 1 2 3 4 5 6 1 2 3 4 5 6 MONTH MONTH

Linear Functions Another solution method is to substitute some x – values into the given function and find y – values to create a table to check points. When 𝑥=0 , 𝑦=4000 0 +4000=4000 𝒙 𝒚 4000 32 32 ( in thousands ) Expenses ( in thousands ) Expenses 28 28 24 24 20 20 16 16 12 12 8 8 4 4 1 2 3 4 5 6 1 2 3 4 5 6 MONTH MONTH

Linear Functions Another solution method is to substitute some x – values into the given function and find y – values to create a table to check points. When 𝑥=0 , 𝑦=4000 0 +4000=4000 When 𝑥=1, 𝑦=4000 1 +4000=8000 𝒙 𝒚 4000 1 8000 32 32 ( in thousands ) Expenses ( in thousands ) Expenses 28 28 24 24 20 20 16 16 12 12 8 8 4 4 1 2 3 4 5 6 1 2 3 4 5 6 MONTH MONTH

Linear Functions Another solution method is to substitute some x – values into the given function and find y – values to create a table to check points. When 𝑥=0 , 𝑦=4000 0 +4000=4000 When 𝑥=1, 𝑦=4000 1 +4000=8000 When 𝑥=2, 𝑦=4000 2 +4000=12000 𝒙 𝒚 4000 1 8000 2 12000 32 32 ( in thousands ) Expenses ( in thousands ) Expenses 28 28 24 24 20 20 16 16 12 12 8 8 4 4 1 2 3 4 5 6 1 2 3 4 5 6 MONTH MONTH

Linear Functions Another solution method is to substitute some x – values into the given function and find y – values to create a table to check points. When 𝑥=0 , 𝑦=4000 0 +4000=4000 When 𝑥=1, 𝑦=4000 1 +4000=8000 When 𝑥=2, 𝑦=4000 2 +4000=12000 𝒙 𝒚 4000 1 8000 2 12000 32 32 ( in thousands ) Expenses ( in thousands ) Expenses 28 28 24 24 20 20 16 16 12 12 8 8 4 4 As you can see, the points are located on the graph on the right. 1 2 3 4 5 6 1 2 3 4 5 6 MONTH MONTH

Linear Functions In some cases, the function will be described verbally and you will have to translate the text into a mathematical statement in the form of a function.

Linear Functions In some cases, the function will be described verbally and you will have to translate the text into a mathematical statement in the form of a function. EXAMPLE : Grant has a base salary of $250 per week selling vacuum cleaners. He makes a $75 commission on all vacuum cleaners he sells during the week. Create a function that shows Grant’s income for each week.

Linear Functions In some cases, the function will be described verbally and you will have to translate the text into a mathematical statement in the form of a function. EXAMPLE : Grant has a base salary of $250 per week selling vacuum cleaners. He makes a $75 commission on all vacuum cleaners he sells during the week. Create a function that shows Grant’s income for each week. A table of values will help us find the function.

Linear Functions In some cases, the function will be described verbally and you will have to translate the text into a mathematical statement in the form of a function. EXAMPLE : Grant has a base salary of $250 per week selling vacuum cleaners. He makes a $75 commission on all vacuum cleaners he sells during the week. Create a function that shows Grant’s income for each week. A table of values will help us find the function. If Grant sells zero vacuum cleaners he gets only his salary. 𝒙 𝒚 250

Linear Functions In some cases, the function will be described verbally and you will have to translate the text into a mathematical statement in the form of a function. EXAMPLE : Grant has a base salary of $250 per week selling vacuum cleaners. He makes a $75 commission on all vacuum cleaners he sells during the week. Create a function that shows Grant’s income for each week. A table of values will help us find the function. If Grant sells zero vacuum cleaners he gets only his salary. If Grant sells one vacuum cleaner, he gets his salary plus $75.00 which is $250 + $75 = $325 𝒙 𝒚 250 1 325

Linear Functions In some cases, the function will be described verbally and you will have to translate the text into a mathematical statement in the form of a function. EXAMPLE : Grant has a base salary of $250 per week selling vacuum cleaners. He makes a $75 commission on all vacuum cleaners he sells during the week. Create a function that shows Grant’s income for each week. A table of values will help us find the function. If Grant sells zero vacuum cleaners he gets only his salary. If Grant sells one vacuum cleaner, he gets his salary plus $75.00 which is $250 + $75 = $325 If Grant sells two vacuum cleaners, he gets his salary plus 75 x 2 which is $250 + $150 = $400 𝒙 𝒚 250 1 325 2 400

Linear Functions In some cases, the function will be described verbally and you will have to translate the text into a mathematical statement in the form of a function. EXAMPLE : Grant has a base salary of $250 per week selling vacuum cleaners. He makes a $75 commission on all vacuum cleaners he sells during the week. Create a function that shows Grant’s income for each week. From the table, our y – intercept is ( 0 , 250 ) 𝒙 𝒚 250 1 325 2 400

Linear Functions In some cases, the function will be described verbally and you will have to translate the text into a mathematical statement in the form of a function. EXAMPLE : Grant has a base salary of $250 per week selling vacuum cleaners. He makes a $75 commission on all vacuum cleaners he sells during the week. Create a function that shows Grant’s income for each week. From the table, our y – intercept is ( 0 , 250 ) Using two points and the slope formula… 𝑚= 400−325 2−1 = 75 1 =75 𝒙 𝒚 250 1 325 2 400

Linear Functions In some cases, the function will be described verbally and you will have to translate the text into a mathematical statement in the form of a function. EXAMPLE : Grant has a base salary of $250 per week selling vacuum cleaners. He makes a $75 commission on all vacuum cleaners he sells during the week. Create a function that shows Grant’s income for each week. From the table, our y – intercept is ( 0 , 250 ) Using two points and the slope formula… 𝑚= 400−325 2−1 = 75 1 =75 Using slope – int. form Grant’s weekly income function is 𝑦=75𝑥+250 𝒙 𝒚 250 1 325 2 400

Linear Functions Whether presented as a table, an equation, or a graph, a linear function has a specific meaning within the context given. Correctly interpreting the proper components of slope and y – intercept is needed.

Linear Functions Whether presented as a table, an equation, or a graph, a linear function has a specific meaning within the context given. Correctly interpreting the proper components of slope and y – intercept is needed. EXAMPLE : An online booking agency charges for tickets and includes a ticketing fee for each order. The total charge, 𝑐 , in dollars, for any number of tickets, 𝑡 , is described by the function 𝑐=20𝑡+4. Which statement is true ? A ) The cost of 20 tickets is $80 B ) The cost of 4 tickets is $20 C ) Each ticket costs $20, and the ticketing fee is $4 D ) Each ticket costs $4, and the ticketing fee is $20

Linear Functions Whether presented as a table, an equation, or a graph, a linear function has a specific meaning within the context given. Correctly interpreting the proper components of slope and y – intercept is needed. EXAMPLE : An online booking agency charges for tickets and includes a ticketing fee for each order. The total charge, 𝑐 , in dollars, for any number of tickets, 𝑡 , is described by the function 𝑐=20𝑡+4. Which statement is true ? A ) The cost of 20 tickets is $80 80≠20 20 +4 𝑤ℎ𝑒𝑛 𝑡=20 B ) The cost of 4 tickets is $20 C ) Each ticket costs $20, and the ticketing fee is $4 D ) Each ticket costs $4, and the ticketing fee is $20 We can rule out A with substitution…

Linear Functions Whether presented as a table, an equation, or a graph, a linear function has a specific meaning within the context given. Correctly interpreting the proper components of slope and y – intercept is needed. EXAMPLE : An online booking agency charges for tickets and includes a ticketing fee for each order. The total charge, 𝑐 , in dollars, for any number of tickets, 𝑡 , is described by the function 𝑐=20𝑡+4. Which statement is true ? A ) The cost of 20 tickets is $80 80≠20 20 +4 𝑤ℎ𝑒𝑛 𝑡=20 B ) The cost of 4 tickets is $20 20≠20 4 +4 𝑤ℎ𝑒𝑛 𝑡=4 C ) Each ticket costs $20, and the ticketing fee is $4 D ) Each ticket costs $4, and the ticketing fee is $20 We can rule out A with substitution… We can rule out B with substitution…

Linear Functions Whether presented as a table, an equation, or a graph, a linear function has a specific meaning within the context given. Correctly interpreting the proper components of slope and y – intercept is needed. EXAMPLE : An online booking agency charges for tickets and includes a ticketing fee for each order. The total charge, 𝑐 , in dollars, for any number of tickets, 𝑡 , is described by the function 𝑐=20𝑡+4. Which statement is true ? A ) The cost of 20 tickets is $80 80≠20 20 +4 𝑤ℎ𝑒𝑛 𝑡=20 B ) The cost of 4 tickets is $20 20≠20 4 +4 𝑤ℎ𝑒𝑛 𝑡=4 C ) Each ticket costs $20, and the ticketing fee is $4 D ) Each ticket costs $4, and the ticketing fee is $20 We can rule out A with substitution… We can rule out B with substitution… C is our answer. Slope describes the cost per ticket, the y – intercept describes the fee.