A linear function is an equation that graphs as a straight line, with the general form of y = mx + b, where m is the slope and b is the y-intercept. A linear equation can be offset by changing the b term, which shifts the line up or down but does not change its slope. The slope of the line can be changed by multiplying the x term by a different value for m. Graphing linear equations with different slopes and offsets demonstrates how varying the terms affects the resulting line.

1. Scaling Analog Data
Introduction to Linear Functions

2. 2
 What is a function?
 A function is nothing more than an equation
that returns a result based on the value of a
variable or variables (tag or tags) within the
equation.
Functions
𝑓 𝑥 = 2𝑥 + 1
X f(x)
-1 -1
0 1
1 3

3. 3
 A linear equation is an equation, that
when graphed, greats a straight line.
 The equation for a straight line is:
Linear Equation
𝑓 𝑥 = 𝑚𝑥 + 𝑏
𝑊ℎ𝑒𝑟𝑒:
𝑚 = 𝑠𝑙𝑜𝑝𝑒
𝑏 = 𝑌 − 𝐼𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡
𝑇ℎ𝑖𝑠 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑐𝑎𝑛 𝑎𝑙𝑠𝑜 𝑏𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑒𝑑 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑎𝑠 :
𝑦 = 𝑚𝑥 + 𝑏

4. 4
 Let’s graph the linear equation:
Graphing a Linear
Equation
𝑦 = 2𝑥 𝑓𝑜𝑟 𝑥 = −5 𝑡𝑜 + 5x y
-5 -10
-4 -8
-3 -6
-2 -4
-1 -2
0 0
1 2
2 4
3 6
4 8
5 10

5. 5
 An offset (the point where the line crosses
the y-axis) can be applied to linear
equation.
 An offset will shift the position of the line
on the graph but will not change the slope
of the line.
𝑦 = 2𝑥 + 2 (𝑓𝑜𝑟 𝑥 = −5 𝑡𝑜 + 5)
 The Y-Intercept is now 2 because b = 2.
The offset is at y = 2, x = 0.
Offset

6. 6
X y = 2x y = 2x + 2
-5 -10 -8
-4 -8 -6
-3 -6 -4
-2 -4 -2
-1 -2 0
0 0 2
1 2 4
2 4 6
3 6 8
4 8 10
5 10 12
Offset
𝑦 = 2𝑥 + 2 (𝑓𝑜𝑟 𝑥 = −5 𝑡𝑜 + 5)

7. 7
Offset
𝑦 = 2𝑥 + 2 (𝑓𝑜𝑟 𝑥 = −5 𝑡𝑜 + 5)

8. 8
 The slope of the line is changed if the x
term is multiplied by a different value.
 The 2 in the previous example is a slope
of 2 or m = 2.
 If the value of the slope (m) changes, the
slope of the line changes.
 The equation:
𝑦 = 4𝑥 (𝑓𝑜𝑟 𝑥 = −5 𝑡𝑜 + 5)
 Will change the slope of the line.
Slope

9. 9
x y = 2x y = 2x + 2 y = 4x
-5 -10 -8 -20
-4 -8 -6 -16
-3 -6 -4 -12
-2 -4 -2 -8
-1 -2 0 -4
0 0 2 0
1 2 4 4
2 4 6 8
3 6 8 12
4 8 10 16
5 10 12 20
𝑦 = 4𝑥 (𝑓𝑜𝑟 𝑥 = −5 𝑡𝑜 + 5)

10. 10
Slope
𝑦 = 4𝑥 (𝑓𝑜𝑟 𝑥 = −5 𝑡𝑜 + 5)