P1.Fungsi_Linear kalkulus 2 aljabar fung

kalkulus

1. Bab-1
Fungsi Linear dan
Fungsi Non Linear

2. Linear functions
06/08/24 copyright 2006 www.brainybetty.com 2

3. A. Linear functions
Equation
Caution: m and b FIXED: parameters
x and y: VARIABLES!
The graph of a linear function with equation y=mx +b is
- a STRAIGHT LINE
3.1 p138)
A function f is a linear function if and only if
f(x) can be written in the form
f(x)=y=mx + b
where m, b are constants.

4. A. Linear functions
Fs : y = 2x + 5 merupakan grs lurus, gambar dicari dengan mengambil
2 titik. Misal x = 0 , y = 5 dan jika x = 1, y = 7, Melalui titik (0,5) dan
(1,7) garis lurusnya bis digambar.
y = 2x + 5

1 5
2
14
x
y

5. A. Linear functions
Significance of the parameter m
•Graphically:
if x is increased by 1 unit,
y is increased by m units
m is the SLOPE of the straight line

6. A. Linear functions
Significance of the parameter m
•Taxi company A: y = 2x + 5, m = 2: the price per
km.
•Numerically: m is CHANGE OF y WHEN x IS
INCREASED BY 1
INPUT OUTPUT
x y
3 11
4 13
x = 1 y = 2
m is the RATE OF CHANGE of
the linear function

7. INPUT OUTPUT
x y
3 11
6 17
•Always:
A. Linear functions
Significance of the parameter m
•Taxi company A: y = 2x + 5, m = 2: the price per
km.
•If x is increased by e.g. 3 (the ride is 3 km longer),
y will be increased by 2  3 = 6 (we have to pay 6
Euro more).
y = mx
(INCREASE FORMULA)
x = 3 y = 3x2=6

8. A. Linear functions
Significance of the parameters b and m
The graph of a linear function with equation
y=mx +b is
- a STRAIGHT LINE
- with y-intercept b
- and slope m
The equation y=mx +b is called the slope-
intercept form of the line with slope m and
intercept b. It is also called an explicit
equation of the line.

9. A. Linear functions
Slope of the line m
Sign of m determines whether the linear function is
- increasing / constant(!!) / decreasing
-Note: what about a vertical line ?
(Section 3.1 p128-129)
-2 2
-2
2
x
y
m < 0
-2 2
-2
2
x
y
m = 0
-2 2
-2
2
x
y
m > 0
(Section 3.1 p131-
Example 6)

10. A. Linear functions
Equation of lines
A straight line through a given point (x
0, y
0)
and
with a given slope m satisfies the equation:
This equation is called the point-
slope form of the line
(Section 3.1 p129-131)
The equation y=mx+b is called the slope-intercept
form of the line
Remember:
 
0 0
y y m x x  
 
0 0
y y m x x  

11. A. Linear functions
Equation of lines
Note that e.g. the vertical line with equation x=2 can
not be written in the slope-intercept form nor in the
slope-point form
(Section 3.1 p129-131)
The equation of a straight line can always be written
using the general linear form Ax+By+C=0 or Ax + By
= C (A and B not both 0).

12. A. Linear functions
Exercises 1.
Tentukan pers. Garis lurus dengan gradien 2 dan
melalui titik (1, -3) dengan
- Point-slope form
- Slope-intercept form
- General linear form

13. Fungsi Linear
Membentuk persamaan fungsi linear :
Melalui 2 titik, yaitu A(x
1,y
1) dan B(x
2,y
2) ?

14. Fungsi Linear (2)
Apabila melalui dua titik:

15. Dua grafik fungsi linear y = a
1
+b
1
x dan
y = a
2 + b
2x, akan :
1.Sejajar (//) jika b
1= b
2 dan
a
1 ≠ a
2`
2. Berpot. tegak lurus (
┴) jika
b
1.b
2=-1
3.Berimpit, dua buah garis akan berimpit
apabila persamaan garis yang satu
merupakan kelipatan dari (proporsional
terhadap) persamaan garis yang lain
Hubungan Dua Garis Lurus

16. Soal persamaan fungsi linear
1.Grs lurus K melalui titik A(60,50) dan grs K
sejajar dgn grafik y = ½ x +790. Grs L
melalui titik B(10,50) dan grs K
berpotongan tegak lurus dgn grs L.
Tentukan persamaan grs K & L
2.Grs lurus p dan q berpotongan tegak lurus
di titik A(10,5). Grs p sejajar dg grafik
fungsi y=- ½x +5. Tentukan bentuk
persamaan grs lurus p dan q.

17. Referensi
•https://byjus.com/maths/linear-functions/
•https://www.cuemath.com/calculus/linear-functions/
•https://www.turito.com/learn/math/linear-functions
•https://physicscatalyst.com/maths/linear-function.php
06/08/24 copyright 2006 www.brainybetty.com 17