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TExES Mathematics 7-12 (235): Practice & Study Guide47 chapters | 389 lessons

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Instructor:
*Michael Eckert*

Michael has a Bachelor's in Environmental Chemistry and Integrative Science. He has extensive experience in working with college academic support services as an instructor of mathematics, physics, chemistry and biology.

Linear and non-linear functions can be used to model and find solutions to a plethora of problems. Examples of non-linear functions may include exponential and quadratic functions.

There are a variety of functions that can be used to solve a host of real-world problems. Function examples range in type and complexity, depending on the type of phenomena they are modeling. For example, a linear correlation may exist between a good or service and its respective price. In this case, a **linear** function may be used to represent this correlation and to predict cost per number of goods.

There also exist problems in science, which requires a more in-depth analysis, in that the function needed to model them are **non-linear**. For example, we may need a parabolic function (quadratic function) to model and solve problems relating to an object's position or displacement with respect to time.

There may exist a certain type of bolt, which is used in the automotive industry to mount engines to cars, that costs exactly 2.50 dollars to fabricate (make). This correlation in price per unit can be described by the equation y = 2.50x, where x = *number of bolts made* and y = *the total cost for x number of bolts made*. Furthermore, a linear regression or line can be made to model this data i.e. *number* vs. *total cost*. The graph of this line will appear as:

We might use the graph to help qualitatively approximate a number vs. total cost or a total cost vs. number. Additionally, we can use its equation y = 2.50x to quantitate specific values along this line. For example, the cost to produce 100 bolts is $250.00.

And the cost of 101, 102, 103, etc. bolts are as follows:

Where x equals 100, 101, 102, 103 and y = 250.00, 252.50, 255.00 and 257.50 respectively.

Non-linear (as well as linear) functions are used to model problems in physical science. One such example are problems associated with an object's motion. For instance, we can model an objects height or vertical displacement through a given time. If a ball is projected from some height, its position through time might be represented by y = -t2 + t + 9, where t = time t in seconds and y = vertical height of the ball in feet. This is an example of a quadratic equation. We can use this function's graph to make qualitative observations e.g. estimating roughly from what points on the graph that the ball was projected out and when it hits the ground or comes to rest.

To make more exact approximations, we can use the equation of y = -t2 + t + 9. For instance, at t = 0 seconds, the ball was projected from y = 9 ft., as y = -02+ 0 + 9 = 9.

We can also determine the exact time that the ball hits the ground at y = 0: 0 = -t2 + t + 9. We can do this by finding the positive root of -t2 + t + 9, noting that we cannot have a negative time.

We find that the positive root is approximately 3.54, so the ball hits the ground at y = 0, at t = 3.54 seconds (at the point where the function of y = -t2 + t + 9 crosses the x-axis).

In this lesson, we illustrated a couple of examples where functions can be used to model and solve for some real-world problems. We looked at a **linear** function and how it related to a problem of positive correlation between units of x (number of bolts) and units of y (cost of number of bolts). Secondly, we looked at a **non-linear** parabolic (quadratic) function, which represented and was used to solve problems regarding the motion of an object through a trajectory or projection.

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TExES Mathematics 7-12 (235): Practice & Study Guide47 chapters | 389 lessons

- What is a Parabola? 4:36
- Parabolas in Standard, Intercept, and Vertex Form 6:15
- How to Factor Quadratic Equations: FOIL in Reverse 8:50
- Factoring Quadratic Equations: Polynomial Problems with a Non-1 Leading Coefficient 7:35
- How to Complete the Square 8:43
- How to Solve a Quadratic Equation by Factoring 7:53
- How to Use the Quadratic Formula to Solve a Quadratic Equation 9:20
- How to Solve Quadratics That Are Not in Standard Form 6:14
- How to Solve a Quadratic Equation by Graphing
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- Using Linear & Quadratic Functions to Problem Solve
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