LINEAR FUNCTIONS
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This document discusses linear functions and straight line graphs. It defines key concepts such as the standard form of a linear equation (y=mx+c), where m is the gradient and c is the y-intercept. It explains how to calculate the gradient between two points and interprets positive and negative gradients. The document also covers finding the x-intercept and y-intercept of a line, and defines the domain and range of linear functions.
![Reference List
1. Barek, N. (2013). Topic 1: Section 1.1 –Graphs of linear equations. Available from Slideshare at
https://www.slideshare.net/nahomyitbarek/topic1-16365779?qid=bf814970-084d-4a05-
a611-e98f6bd2dfad&v=&b=&from_search=10 [accessed on 21 August 2022].
2. Estelav, (2013). Linear Functions. Available from Slideshare at
https://www.slideshare.net/estelav/linear-functions-20097351?qid=9219ab42-b21c-4238-
bd52-b3ec7a683dc6&v=&b=&from_search=4 [Accessed on 21 August 2022].
3. Geraldine, M. (2021). Graph of linear function. Available from Slideshare at
https://www.slideshare.net/MartinGeraldine/graph-of-linear-function [Accessed on 20 August
2022].
4. MacLane, S., (2012). Mathematics form and function. Springer Science & Business Media.
5. Mushipe, M. and Ogbonnaya, U. I., (2019). Geogebra and Grade 9 learners’ achievement in linear
functions.
6. Racso, E. (2014). AS LEVEL Function (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMS.
Available from Slideshare at https://www.slideshare.net/RACSOstudentHELP/function-
edited?qid=20b4981c-afc2-4e55-a270-340a9f5c616f&v=&b=&from_search=10 [accessed on 20
August 2022].
7. Teacher Andoo, (2012). Linear functions 1. Available from Slideshare at
https://www.slideshare.net/teacherandoo/linear-functions-1?qid=7eb23bb1-4f42-42cf-ab16-
abe868bbf7af&v=&b=&from_search=2 [Accessed on 21 August 2022].
8. Volunteers, Siyavula Grade 10 Mathematics. Department of Basic Education. South Africa.](https://image.slidesharecdn.com/nkadimengmt219015363linearfunctionsslidesharere-purposedslidesppt-220821153454-e680b5a4/85/LINEAR-FUNCTIONS-12-320.jpg)
LINEAR FUNCTIONS
- 1. Linear functions. STRAIGHT LINE GRAPH.
- 2. RECAP!!! SPIDER WEB. In a spider web, there are x values, y values and also a rule. On your left, this is where your x- values(input) lie. Therefore, your y- values(output) are on the right hand side. Lastly, the rule is always in the middles. The x- and y-values are there to help one plot a graph on a cartesian plane. A cartesian plane consist of all the integers in the number system.
- 3. Drawing a graph. Rule: 3x-1 x -3 -2 -1 0 1 2 3 y -10 -7 -4 -1 2 5 8
- 4. Sketching a straight line graph
- 5. What is a function? A function is an expression or rule that defines a relationship between a variable that is dependent and a variable that is independent. In a cartesian plane, the independent variable are the x-values, the dependent variable are the y-values. The y-values lie on the y axis, and the x-values lie on the x axis (Barek, 2013). The standard form/equation of a straight line graph is defined as: 𝑦 = 𝑚𝑥 + 𝑐 Where M is the gradient (change in y over the change in x), and c is the position above or below the x axis. The gradient M, is the change in y over the change in x. (such that 𝑀 = Δ𝑦 Δ𝑥 = 𝑦1−𝑦2 𝑥1−𝑥2 ). Example 1: Calculate the gradient between points A(3,6) and B(4, 9). Solution: M(AB)= 𝑌𝐴−𝑌𝐵 𝑥𝐴−𝑥𝐵 = 6−9 3−4 = −3 −1 = 3.
- 6. POSITIVE OR NEGATIVE GRADIENT. Since the above example has a positive gradient, this means that the graph will be increasing from - ∞ 𝑡𝑜 ∞. If the gradient was negative, therefore the graph will be decreasing from -∞ 𝑡𝑜 ∞. The graph will look like this:
- 7. Gradient, continued… https://www.youtube.com /watch?v=T3n74ul5-o8 https://www.youtube.com /watch?v=MYCH7gswI4k https://www.mashupmath .com/blog/finding-slope- of-a-line-rise-over-run
- 8. The effect of “c” The value of affects the graph when it cuts the y axis. This is also known the y intercept of the graph. Therefore, if c>0, the graph shift vertically upwards. And if c<0, the graph shifts vertically downwards. M<0 C>0 C<0 M>0
- 10. The intercepts of a straight line graph. The intercepts of a graph are points that cuts either the y axis or the x axis that lies on the graph. The x- and y-intercepts are used in sketching linear functions. Because we only need two points to graph a line, the intercepts are all we require and the simplest coordinates to get. To obtain the two points, we will set the values of x and y to 0 (Geraldine, 2021). To calculate the x intercept, let the y value be equated to zero. Example 2: 𝑦 = 3𝑥 − 1. let y=0 0 = 3𝑥 − 1 1 = 3𝑥 𝑥 = 1 3 . The graph cuts at ( 1 3 , 0) on the x axis. To calculate the y intercept, let the x value be equated to zero. Therefore, in this case, we substitute the variable x with zero. 𝑦 = 3𝑥 − 1 = 3 0 − 1 = −1 The graph cuts at (0, -1) on the y axis.
- 11. Domain and Range DOMAIN: this is where the x values are defined in the straight line graph. RANGE: this is where the y values are defined in a straight line graph. The domain for a straight line graph ( y=mx+c) is {𝑥: 𝑥 ∈ 𝑅} because there is no value of x for which y is undefined. The range of the straight line graph (y=mx+c) is {y: 𝑦 ∈ 𝑅} because y can take on any real value/number.
- 12. Reference List 1. Barek, N. (2013). Topic 1: Section 1.1 –Graphs of linear equations. Available from Slideshare at https://www.slideshare.net/nahomyitbarek/topic1-16365779?qid=bf814970-084d-4a05- a611-e98f6bd2dfad&v=&b=&from_search=10 [accessed on 21 August 2022]. 2. Estelav, (2013). Linear Functions. Available from Slideshare at https://www.slideshare.net/estelav/linear-functions-20097351?qid=9219ab42-b21c-4238- bd52-b3ec7a683dc6&v=&b=&from_search=4 [Accessed on 21 August 2022]. 3. Geraldine, M. (2021). Graph of linear function. Available from Slideshare at https://www.slideshare.net/MartinGeraldine/graph-of-linear-function [Accessed on 20 August 2022]. 4. MacLane, S., (2012). Mathematics form and function. Springer Science & Business Media. 5. Mushipe, M. and Ogbonnaya, U. I., (2019). Geogebra and Grade 9 learners’ achievement in linear functions. 6. Racso, E. (2014). AS LEVEL Function (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMS. Available from Slideshare at https://www.slideshare.net/RACSOstudentHELP/function- edited?qid=20b4981c-afc2-4e55-a270-340a9f5c616f&v=&b=&from_search=10 [accessed on 20 August 2022]. 7. Teacher Andoo, (2012). Linear functions 1. Available from Slideshare at https://www.slideshare.net/teacherandoo/linear-functions-1?qid=7eb23bb1-4f42-42cf-ab16- abe868bbf7af&v=&b=&from_search=2 [Accessed on 21 August 2022]. 8. Volunteers, Siyavula Grade 10 Mathematics. Department of Basic Education. South Africa.