### “The height of a plant that starts at 40 cm and grows 1 cm per week.”

#### Part 1:

This snap I specifically chosen interested me by its steady growth pattern and it’s simple starting height. Its steady growth pattern has been consistent for the past 2 weeks even though plants usually grow at uncontrolled rates making it unique. Also its starting height,which is 40 cm, is considerably large and unique compared to my other snaps of linear relations.

This partial linear relation, has an Independent Variable and a Dependent Variable. The Independent Variable in the relation is 1 cm increase per week or x is measured by calculating its new height each week, the week after, and so on.The dependent variable in the relation is the total height of the plant or y depends on the initial value and the slope. It cannot be determined if the other other variables are not found.

### Reflection and Learning Goals

**Linear Relations**

Using Data Management to Investigate Relationships // Understanding Characteristics of Linear Relations // Connecting Various Representations of Linear Relations

**Analytic Geometry**

Investigating the Relationship Between the Equation of a Relation and the Shape of Its Graph // Investigating the Properties of Slope // Using the Properties of Linear Relations to Solve Problems

My linear relation reflects on these learning goals as this linear relation broadly describes its connecting topics like T.O.V, slope y-intercept form, standard form, P.O.I, and more. Also textbook questions and examples helped explain the connections with linear relations as it reviews the learning goals explicitly.

### Part II:

#### Phase I:

In “The height of a plant that starts at 40 cm and grows 1 cm per week.”, Let ‘y’ be the total height of the plant and let ‘x’ be the amount of weeks the plant has gone through.

y = x +40 was found by first find its starting height to find b. Then find the speed by calculating how much it grew per week to found m.

The linear relation in slope y-intercept form is y = x + 40 (y=mx+b) where ‘m’ represents the speed the plant is growing which is 1. Also where ‘b’ represents the original starting height of the plant before it was measured and snapped.

In standard form, the equation is x - y + 40 = 0. The x-intercept is where the graph intercepts the x-axis and the y-intercept is where the graph intercepts the y-axis. Algebraically, the x-intercept would be x = 0 and the y-intercept would be y = 0.

#### Phase II:

**y = x + 40** to** y = 41** where the m value was decreased by 1 to 0 and the b value was increased by 1 to 41. This new equation gave a slope of 0 and means the plant height (y) will stay the same as the initial value over the several weeks. Another difference is the increase in initial value from 40 to 41.

If the m value was increased and the initial value decreased, the new equation would be y = 2x +39. The growth of the plant (m) will drastically increase as it slope is double the speed of the original equation. Also the initial value has decreased giving it a slightly smaller height than the original but then compensates with its new speed. This new slope may happen as the height of the plant is unpredictable but is constant.

With the new equations, **y = 41** and **y = 2x + 39**, and the original **y = x + 40**, all their Point of Intersections (P.O.I.) are at **(1, 41)**

Credits:

Created with images by Florian Schwalsberger - "trees"