4.3 Linear Functions materials grade 8.pptx

4.3 Linear Functions materials grade 8.pptx

1. MATHEMATICS 8 Linear Functions

2. Learning Competency Graphs and illustrates a linear function and its : (a) domain (b) r ange (c) table of values; (d) intercepts; and (e) slope 2

3. Linear Function 3 A linear function is a function that can be written in the form 𝒇 𝒙 = 𝒎𝒙 + 𝒃 ✘ where 𝑚 and 𝑏 are real numbers ✘ where 𝑚 tells us the slope of a line and 𝑏 tells us where the graph crosses the 𝑦 - axis.

4. Linear Function 4 𝒇 𝒙 = 𝒎𝒙 + 𝒃 A linear function is a function whose graph is a straight line. Its equation can be written in the form 𝒚 = 𝒎𝒙 + 𝒃 , where 𝑥 and 𝑦 are used for the independent and dependent variables, respectively, 𝑚 ≠ .

5. Linear Function 5 𝒚 = 𝒇 𝒙 𝒇 𝒙 = 𝒎𝒙 + 𝒃 𝒚 = 𝒎𝒙 + 𝒃 ✘ 𝑓(𝑥) means “the value of 𝑓 at 𝑥 ” 𝒈(𝒙) or 𝒉(𝒙) ✘ letters other than 𝑓 such as 𝐺 and 𝐻 , or 𝑔 and ℎ can also be used.

6. 6 Linear Function as an Equation Which of the following function is linear? 𝒚 = 𝒎𝒙 + 𝒃 𝒇 𝒙 = 𝒎𝒙 + 𝒃 1. 𝑦 = 8𝑥 − 5 2. 𝑦 = − 𝑥 3 + 2 3. 𝑦 = 𝑥 𝑥 4. 𝑦 = 2 + 7 5. 𝑦 = 𝑥 2 − 3 Linear Function. It is in the form 𝑦 = 𝑚𝑥 + 𝑏 , where 𝑚 = 8 and 𝑏 = −5 Linear Function. By rewriting the equation, we can have 𝑦 = 1 1 3 3 − 𝑥 + 2 where 𝑚 = − and 𝑏 = 2 . Linear Function. By rewriting the equation, we can have 𝑦 = 𝑥 + where 𝑚 = 1 and 𝑏 = . Not a Linear Function. It cannot be expressed in the form 𝑦 = 𝑚𝑥 + 𝑏 because 𝑥 is in the denominator. Not a Linear Function. The degree of the equation is on the second degree.

7. A linear function can also be described using its graph . Let’s determine the values of the function 𝑓 if 𝑓 𝑥 = 2x + 1 at 𝑥 = − 2, −1, 0, 1, and 2 , and see if it will illustrate a straight line. SOLUTION: = 2𝑥 + 1 = 2 −2 + 1 1. 𝑓 𝑥 𝑓 −2 𝑓 −2 𝒇 −𝟐 = −4 + 1 = −𝟑 𝒙 𝒇(𝒙) −𝟐 −𝟑 Ordered pair: (−𝟐, −𝟑) REMEMBER: Note that an ordered pair (𝑥, 𝑦) can be written as (𝑥, 𝑓 𝑥 ) for any function in 𝑓 𝑥 notation. 7

8. A linear function can also be described using its graph . Let’s determine the values of the function 𝑓 if 𝑓 𝑥 = 2x + 1 at 𝑥 = − 2, −1, 0, 1, and 2 , and see if it will illustrate a straight line. SOLUTION: = 2𝑥 + 1 = 2 −1 + 1 2. 𝑓 𝑥 𝑓 −1 𝑓 −1 𝒇 −𝟏 8 = −2 + 1 = −𝟏 Ordered pair: (−𝟏, −𝟏) 𝒙 𝒇(𝒙) −𝟐 −𝟑 −𝟏 −𝟏

9. A linear function can also be described using its graph . Let’s determine the values of the function 𝑓 if 𝑓 𝑥 = 2x + 1 at 𝑥 = − 2, −1, 0, 1, and 2 , and see if it will illustrate a straight line. SOLUTION: 3. 𝑓 𝑥 = 2𝑥 + 1 𝑓 = 2 + 1 𝑓 = 0 + 1 𝒇 𝟎 = 𝟏 Ordered pair: (𝟎, 𝟏) 9 𝒙 𝒇(𝒙) −𝟐 −𝟑 −𝟏 −𝟏 𝟎 𝟏

10. A linear function can also be described using its graph . Let’s determine the values of the function 𝑓 if 𝑓 𝑥 = 2x + 1 at 𝑥 = − 2, −1, 0, 1, and 2 , and see if it will illustrate a straight line. SOLUTION: 4. 𝑓 𝑥 = 2𝑥 + 1 𝑓 1 = 2 1 + 1 𝑓 1 = 2 + 1 𝒇 𝟏 = 𝟑 Ordered pair: (𝟏, 𝟑) 10 𝒙 𝒇(𝒙) −𝟐 −𝟑 −𝟏 −𝟏 𝟎 𝟏 𝟏 𝟑

11. A linear function can also be described using its graph . Let’s determine the values of the function 𝑓 if 𝑓 𝑥 = 2x + 1 at 𝑥 = − 2, −1, 0, 1, and 2 , and see if it will illustrate a straight line. SOLUTION: 5. 𝑓 𝑥 = 2𝑥 + 1 𝑓 2 = 2 2 + 1 𝑓 2 = 4 + 1 𝒇 𝟐 = 𝟓 Ordered pair: (𝟐, 𝟓) 11 𝒙 𝒇(𝒙) −𝟐 −𝟑 −𝟏 −𝟏 𝟎 𝟏 𝟏 𝟑 𝟐 𝟓

12. A linear function can also be described using its graph . Let’s determine the values of the function 𝑓 if 𝑓 𝑥 = 2x + 1 at 𝑥 = −2, −1, 0, 1, and 2 , and see if it will illustrate a straight line. 𝒙 𝒇(𝒙) −𝟐 −𝟑 −𝟏 −𝟏 𝟎 𝟏 𝟏 𝟑 𝟐 𝟓 12

13. 13 A linear function can also be illustrated using a table of values . 𝒙 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝒇(𝒙) −3 −1 1 3 5 We can do this by looking at the first difference of the 𝑥 - coordinates and 𝑦 - coordinates. +𝟏 +𝟏 +𝟏 +𝟏 +𝟐 +𝟐 +𝟐 +𝟐 Since both quantities change by constant amounts, this means that the relationship between the quantities is linear .

14. A linear function can also be illustrated using a table of values . 𝒙 𝟏 𝟐 𝟑 𝟒 𝟓 𝒇(𝒙) 3 6 9 12 15 +𝟏 +𝟏 +𝟏 +𝟏 +𝟑 14 +𝟑 +𝟑 +𝟑 Since the 𝑥 -coordinates and the 𝑦 - coordinates increase by constant amounts, this table of values illustrates a linear function .

15. A linear function can also be illustrated using a table of values . 𝒙 −𝟔 −𝟒 −𝟐 𝟎 𝟐 𝒇(𝒙) 2 3 5 8 12 +𝟐 +𝟐 +𝟐 +𝟐 +𝟑 15 +𝟒 +𝟏 +𝟐 Since the 𝑥 -coordinates and the 𝑦 -coordinates do not increase by constant amounts, this table of values does not illustrate a linear function .

16. A linear function can also be illustrated using a table of values . 𝒙 𝟓 𝟏𝟎 𝟏𝟓 𝟐𝟎 𝟐𝟓 𝒇(𝒙) 25 50 75 100 125 +𝟓 +𝟓 +𝟓 +𝟓 +𝟐𝟓 16 +𝟐𝟓 +𝟐𝟓 +𝟐𝟓 Since the 𝑥 -coordinates and the 𝑦 - coordinates increase by constant amounts, this table of values illustrates a linear function .

17. MATHEMATICS 8 Thank you!