Answer:
See Explanation.
General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Equality Properties
- Multiplication Property of Equality
- Division Property of Equality
- Addition Property of Equality
- Subtract Property of Equality
<u>Algebra I</u>
Slope Formula: ![\displaystyle m=\frac{y_2-y_1}{x_2-x_1}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20m%3D%5Cfrac%7By_2-y_1%7D%7Bx_2-x_1%7D)
Slope-Intercept Form: y = mx + b
Linear Regression
Step-by-step explanation:
We can draw any best line of fit, as long as it is <em>reasonable</em> around the points that are given.
We can just take 2 points and use slope formula and Slope-Intercept Form to find the equation for the best line of fit.
Using Linear Regression, we can determine the <em>true</em> best line of fit using graphing utilities.
<u>Finding the best line of fit</u>
<em>Define 2 points</em>
Point (21600, 205)
Point (27000, 290)
<em>Find slope m</em>
- Substitute in point [SF]:
![\displaystyle m=\frac{290-205}{27000-21600}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20m%3D%5Cfrac%7B290-205%7D%7B27000-21600%7D)
- [Fraction] Subtract:
![\displaystyle m=\frac{85}{5400}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20m%3D%5Cfrac%7B85%7D%7B5400%7D)
- [Fraction] Simplify:
![\displaystyle m=\frac{17}{1080}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20m%3D%5Cfrac%7B17%7D%7B1080%7D)
<em>Find equation</em>
- Define equation [SIF]:
![\displaystyle y = \frac{17}{1080}x + b](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B17%7D%7B1080%7Dx%20%2B%20b)
- Substitute in point:
![\displaystyle 290 = \frac{17}{1080}(27000) + b](https://tex.z-dn.net/?f=%5Cdisplaystyle%20290%20%3D%20%5Cfrac%7B17%7D%7B1080%7D%2827000%29%20%2B%20b)
- Multiply:
![\displaystyle 290 = 425 + b](https://tex.z-dn.net/?f=%5Cdisplaystyle%20290%20%3D%20425%20%2B%20b)
- Isolate y-intercept <em>b</em>:
![\displaystyle -135 = b](https://tex.z-dn.net/?f=%5Cdisplaystyle%20-135%20%3D%20b)
- Rewrite:
![\displaystyle b = -135](https://tex.z-dn.net/?f=%5Cdisplaystyle%20b%20%3D%20-135)
- Redefine equation:
![\displaystyle y = \frac{17}{1080}x - 135](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B17%7D%7B1080%7Dx%20-%20135)
Slope-Intercept Form tells us that our slope <em>m</em> =
and our y-intercept
.
Setting this as function f(x), we can see from the graph that it is extremely accurate (Blue line).
<u>Using Linear Regression</u>
Depending on the graphing calc you have, the steps may be different.
Using a graphing calc, we can use statistics and determine the <em>best</em> best line of fit.
When we determine the values, we should see that our equation would be g(x) (Green Line).
<em>Credit to Lauren for collabing w/ me in graphing.</em>