Homework 2 Assignment

1. Read through sections 1.4 and 1.5 using the study guide below. Section 1.4 should all be review from Algebra 2, so just make sure you understand the objectives below. Section 1.5 will contain some new material.
2. Watch the video for Section 1.5.
3. Complete the homework assignment on course compass.
4. Complete the assigment at the bottom of the study guide. Email or put in my box.

Study Guide

Section 1.4: Equations of Lines

• Lines can be written in Slope- Intercept form or Point-Slope Form.
  • Slope-intercept form is f(x) = mx + b, where m is the slope of the line and b is the y-intercept (the place where the line crosses the y-axis.
    • See examples 2, 3, and 4.
  • Point-Slope form is y – y1 = m (x – x1), where m is the slope of the line and (x1, y1) is any point on the line.
    • See example 6.
• Two lines are Parallel if they have the same slope and different y-intercepts (they change at the same rate).
• Two lines are Perpendicular if the product of their slopes is -1.
  • For example, if you have two lines with slopes of -2 and 0.5, they would be perpendicular.
  • Lines are also perpendicular if one is vertical and the other is horizontal.
  • See examples 7 & 8

Section 1.5: More on Functions

Increasing and Decreasing Functions (see example 1)

• A function is increasing if it is rising from left to right. Mathematically, we say it this way:
  • A function is increasing on an open interval if for all a and b in the interval, a < b implies f(a) < f(b). In other words, if the x-value is smaller, then so is the y-value.
• Similarly, a function is decreasing if it is dropping from right to left. Mathematically, we say it like this:
  • A function is decreasing on an open interval if for all a and b in the interval,a < b implies f(a) > f(b). So if the x-value is larger, then the y-value is smaller.
• A function is constant if the values stay the same from left to right. Mathematically:
  • A function is constant on an open interval if for all a and b in the interval, f(a) = f(b). In other words, every y-value is the same in the interval.

Relative Maxima/ Minima

• We call the hills and valleys in graphs local, or relative, maxima and minima.
• The value of the maximum or minimum is the y-value, or function value. We say that the max or min occurs at the x-value.
  • For example, in the graph below, the local max is 4 and occurs at x = -1. The local min occurs at x = 1 and is equal to -4.
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Piecewise-Defined Functions

A piecewise function uses different output formulas for different parts of the domain. So, for example, the function could be defined by one equation for x < 0, and by another equation for x ≥ 0. See examples 5 – 7 in your textbook.

Assignment:

1. Make up a real-world piecewise defined function that increases, decreases, and is constant (not neccesarily in that order) for different parts of it’s domain.
2. Draw a graph of the function and explain in terms of the domain where the function increases, decreases, and is constant.
3. For a bonus point, write the function as a piecewise defined function with the appropriate equations.
4. Email to me (I made the diagram below just using Paint) or write out by hand and leave in my box on Wed/Thurs.

Example: T(t) is a piecewise function for Temperature as a function of time. It is defined as follows: Water is heated to boiling to make tea. The water boils for a minute and then it is taken off the heat and left on the counter to cool. A graph of the function could look like:

 

 

 

 

 

 

 

 

The function is increasing on t = (0, 2), constant on t = (2, 3) and decreasing on t > 3.