Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Compare the graphs of linear functions with differen..."

2020-06-01T14:40:03Z

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[[1.4 Finding Linear Functions | Prior Lesson]] | [[1.6 Function Notation Input & Output | Next Lesson]]

==Objectives:==
*Compare the graphs of linear functions with different slopes and y-intercepts
*Given a line, find the slopes of a parallel line and a perpendicular line
*Write equations for vertical and horizontal lines

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;Definitions: perpendicular, parallel, same line

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==Lesson Guide==

===Warm-Up===
Have students do Problems 1 and 2.

One way to approach this activity is to have students try to work through it on their own for 5 minutes, and then spend 10 minutes as a class going through the graph. As you go through the questions collaborate with the students to highlight the following facts:

* Decreasing lines have a negative slope and increasing lines have a positive slope. Remember, we ``read" a graph from left to right. So <math>C</math>, <math>D</math> and <math>E</math> are increasing functions, <math>B</math> is a decreasing function and <math>A</math> is neither increasing nor decreasing.
*The ``steepness" of a graph comes from the magnitude, not the ``positiveness" of the slope. So <math>B: y = 2 -x</math> and <math>D: y = x-2</math> have the same steepness even though one has negative and the other positive slope.
*In fact, <math>B</math> and <math>D</math> are perpendicular lines. How do we know? We can tell because <math></math>\text{slope}(D) \times \text{slope}(B) = 1 \times -1= -1<math></math> Note that students (and the text) may also use the term "negative reciprocals." Point out that these are equivalent.
*Horizontal lines have a slope of zero. So for horizontal lines <math>y = mx + b</math> becomes <math>y= b</math>. The equation for <math>A</math> is therefore <math>y = 5</math>
*The <math>y</math>-intercept is just the point where the graph crosses the <math>y</math>-axis. What are the <math>y</math>-intercepts for <math>A</math>, <math>B</math>, <math>D</math> and <math>E</math>? <math>A: (0,5)</math>, <math>B: (0,2)</math>, <math>D \& E: (0,-2)</math>.
'''You might even choose to write some of these facts on the board!''' If possible it is good to let students make the observations and then write them on the board as students list these facts. If students can create the table with your prodding they will be much better off.

====Understand how changing the slope and <math>y</math>-intercept changes the graph of a linear equation====

Use Problem 1 to discuss how the slope reflects whether the graph of the linear function is increasing or decreasing.

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If (a function defined by) <math>y=mx+b</math> is an equation for a line with slope <math>m</math> and <math>y</math>-intercept <math>(0,b)</math>, then
*<math>m>0</math> means that the function is '''increasing''',
*<math>m<0</math> means that the function is '''decreasing''',
*<math>m=0</math> means that the graph of the function is the horizontal line with equation <math>y=b</math>, and
*the larger the magnitude of <math>m</math>, the steeper its graph is.
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====Understand the relationship between slopes of parallel and perpendicular lines====

This section focuses on the relationship between two lines, specifically the relationship between their slopes. Ask students the various relationships that two lines could have (they intersect at a point, they never intersect, or they are the same line). Give a few examples of graphs of pairs of lines and ask students what the relationship is between the lines.

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Once you have done the examples, ask how we could determine the relationship by simply looking at the linear equations of the lines.
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Let line 1 be given by the formula <math>y = m_1x + b_1</math> and let line 2 be given by the formula <math>y = m_2x + b_2</math>. Then, we say that
*lines 1 and 2 are \underline{perpendicular} if <math>m_1 = \frac{-1}{m_2}</math> (negative reciprocal).
*lines 1 and 2 are \underline{parallel} if <math>m_1 = m_2</math> and <math>b_1\neq b_2</math>. (Note: some homework problems may require that <math>b_1\neq b_2</math> but for exams we will make a point that <math>m_1 = m_2</math>. There are two camps on whether it is necessary that <math>b_1\neq b_2</math> and we will try to avoid taking a side.)
*lines 1 and 2 are the \underline{same line} if <math>m_1=m_2</math> {\em and} <math>b_1=b_2.</math>
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The statement <math>m_1 = \frac{-1}{m_2}</math> may confuse students, so saying ``negative reciprocal" may actually be more helpful to them.

Have students do Problem 3(a-g).

====Write equations for vertical and horizontal lines====

Remind students that vertical lines have a fixed <math>x</math>-coordinate for every point on the line. So, we describe a vertical line by simply writing <math>x=c</math> for whatever that constant <math>x</math>-coordinate <math>c</math> is.
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For any constant <math>k</math>:

*The graph of the equation <math>y=k</math> is a horizontal line through <math>(0,k)</math> and its slope is zero.

*The graph of the equation <math>x=k</math> is a vertical line through <math>(k,0)</math> and its slope is undefined.

Have students finish Problem 3.

Tell students that Problem 4 has shown up on many of the exams and they should make sure they have that mastered.

Have students finish Problem 5. Use a graph to show students how the word problem and graph are tied together.

==Comments==

*Students have not actually been taught the meaning of the word magnitude in 100A, so it would probably be good to define magnitude for them.

*I think problem 3 should really be split up. Even in the lesson plan, we don't do the whole problem at once, so I don't see a reason why it should all be one problem. It's just a lot for students to look at and could be overwhelming.

*Regarding #3, I like to assign each group 2-3 parts to do and write up on the board. That way it's not as overwhelming. -Juliana

I have found that stating that perpendicular lines have the negative reciprocal of each other has usually confused students more. It has often helped to actually to present m_1 times m_2 = -1. And then have them solve the equation. Last semester most students forgot the negative with the reciprocal.

1.5 Comparing Linear Functions - Revision history

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