Math 8 - Linear Functions
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This document discusses linear functions and how to represent them using equations, graphs, and tables of values. It defines a linear function as one that can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept. Examples are provided to illustrate determining the slope and y-intercept from an equation and representing a linear function using an equation, table of values, or graph. It is explained that a linear function will produce a straight line on a graph and have constant differences in x- and y-coordinates in its table of values.
Math 8 - Linear Functions
- 1. MATHEMATICS 8 Quarter 2 Week 4 Linear Functions MR. CARLO JUSTINO J. LUNA MALABANIAS INTEGRATED SCHOOL Angeles City
- 2. Learning Competency 2 ✘ Graphs and illustrates a linear function and its (a) domain; (b) range; (c) table of values; (d) intercepts; and (e) slope (M8AL-IId-1)
- 3. Linear Function 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. 3
- 4. Linear Function 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 ≠ 0. 4
- 5. Linear Function 𝒇 𝒙 = 𝒎𝒙 + 𝒃 𝒚 = 𝒎𝒙 + 𝒃 𝒚 = 𝒇 𝒙 𝒈(𝒙) or 𝒉(𝒙) 5 ✘ 𝑓(𝑥) means “the value of 𝑓 at 𝑥” ✘ letters other than 𝑓 such as 𝐺 and 𝐻, or 𝑔 and ℎ can also be used.
- 6. 6 Linear Functionas 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 4. 𝑦 = 2 𝑥 + 7 5. 𝑦 = 𝑥2 − 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 3 𝑥 + 2 where 𝑚 = − 1 3 and 𝑏 = 2. Linear Function. By rewriting the equation, we can have 𝑦 = 𝑥 + 0 where 𝑚 = 1 and 𝑏 = 0. 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. 7 A linear function can alsobe describedusing 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: 1. 𝑓 𝑥 = 2𝑥 + 1 𝑓 −2 = 2 −2 + 1 𝑓 −2 = −4 + 1 𝒇 −𝟐 = −𝟑 Ordered pair: (−𝟐, −𝟑) 𝒙 𝒇(𝒙) −𝟐 −𝟑 REMEMBER: Note that an ordered pair (𝑥, 𝑦) can be written as (𝑥, 𝑓 𝑥 ) for any function in 𝑓 𝑥 notation.
- 8. 8 A linear function can alsobe describedusing 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: 2. 𝑓 𝑥 = 2𝑥 + 1 𝑓 −1 = 2 −1 + 1 𝑓 −1 = −2 + 1 𝒇 −𝟏 = −𝟏 Ordered pair: (−𝟏, −𝟏) 𝒙 𝒇(𝒙) −𝟐 −𝟑 −𝟏 −𝟏
- 9. 9 A linear function can alsobe describedusing 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 0 = 2 0 + 1 𝑓 0 = 0 + 1 𝒇 𝟎 = 𝟏 Ordered pair: (𝟎, 𝟏) 𝒙 𝒇(𝒙) −𝟐 −𝟑 −𝟏 −𝟏 𝟎 𝟏
- 10. 10 A linear function can alsobe describedusing 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) 𝒙 𝒇(𝒙) −𝟐 −𝟑 −𝟏 −𝟏 𝟎 𝟏 𝟏 𝟑
- 11. 11 A linear function can alsobe describedusing 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) 𝒙 𝒇(𝒙) −𝟐 −𝟑 −𝟏 −𝟏 𝟎 𝟏 𝟏 𝟑 𝟐 𝟓
- 12. 12 A linear function can alsobe describedusing 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
- 13. 13 A linear function can alsobe illustratedusinga table of values. We can do this by looking at the first difference of the 𝑥- coordinates and 𝑦-coordinates. 𝒙 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝒇(𝒙) −3 −1 1 3 5 +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. 14 A linear function can alsobe illustratedusinga table of values. 𝒙 𝟏 𝟐 𝟑 𝟒 𝟓 𝒇(𝒙) 3 6 9 12 15 +𝟏 +𝟏 +𝟏 +𝟏 +𝟑 +𝟑 +𝟑 +𝟑 Since the 𝑥-coordinates and the 𝑦-coordinates increase by constant amounts, this table of values illustrates a linear function.
- 15. 15 A linear function can alsobe illustratedusinga table of values. 𝒙 −𝟔 −𝟒 −𝟐 𝟎 𝟐 𝒇(𝒙) 2 3 5 8 12 +𝟐 +𝟐 +𝟐 +𝟐 +𝟏 +𝟐 +𝟑 +𝟒 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. 16 A linear function can alsobe illustratedusinga table of values. 𝒙 𝟓 𝟏𝟎 𝟏𝟓 𝟐𝟎 𝟐𝟓 𝒇(𝒙) 25 50 75 100 125 +𝟓 +𝟓 +𝟓 +𝟓 +𝟐𝟓 +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 Quarter 2 Week 4 Thank you! MR. CARLO JUSTINO J. LUNA MALABANIAS INTEGRATED SCHOOL Angeles City