Math 10C

Notes Chapter 6

Section 6.1 Slope of a Line Math 10C

The SLOPE of a line is the ratio (fancy word for fraction) of the RISE of the line and the RUN of the line. You actually see and use slope in many capacities outside of the classroom:

a. Slope of your patio – does the water run towards your house when it rains? This is BAD

b. The slope of your roof – is the slope so small that you have to get up on the roof to shovel off the snow or is it high enough that heavy snow slides off?

c. The slope of your windshield – is it high enough that all rocks create a chip when it hits or is it low enough that the rock just slides up?

Although we see slope in many “world wide” capacities, we will stick to the math first and then work our way through real life situations.

The slope is a relatively value to determine. In math we use the variable “m” to illustrate slope. There are a few formulas to help you with this calculation

m= riserun

m= ΔyΔx

m=y2− y1x2−x1

There are 4 different types of slopes:

Ultimately there are 3 types of questions that I can ask you:

1. If I give you the graph, can you determine the slope?2. If I give you the slope, can you create a graph that shows it?3. If I give you 2 points, can you determine the slope?

Determine from the GRAPH

Given Slope, Draw and label a graph

Slope=75

Slope=−38

Slope=49 Slope=−8

3

Homework from Calendar

Section 6.2 Slopes of Parallel and Perpendicular Lines Math 10C

Simply stated, lines are parallel if they have THE SAME SLOPE. Lines are perpendicular if they have the NEGATIVE RECIPROCAL SLOPE. Now, many of you will ask, what the heck is the negative reciprocal? It is the negative of the “flip” of the slope of the line in question.

Original Negative Recip.

4 −14

−23

32

−15 5

Are these lines parallel? – the only way to tell for sure is to determine their slopes.

Which lines are perpendicular?

What if you are not given a graph, but are given points? You simply use your formula for slope.

m=y2− y1x2−x1

Line PQ passes through P(-7, 2) and Q(-2, 10) and line RS passes through R(-3, -4) and S(5, 1); are these lines perpendicular, parallel or neither?

Line ST passes through S(-2, 7) and T(2, -5) and line UV passes through U (-.2, 3) and V(7, 6); are these lines parallel, perpendicular or neither?

Homework from Calendar Quiz tomorrow

Section 6.4 Slope Intercept Form Math 10C

The equation of a linear function can be written in the form:

y = mx + b slope y-intercept

Determine the equation if:

a. If the slope is −73 and the y-intercept is 3

______________________b. If the slope is 3 and the y-intercept is -2 ______________________

c. If the slope is 0 and the y-intercept is 7 ______________________

In this form, it is XTREMELY easy to graph a line from this equation…..

Graph y = 35x + 3 Graph y=

−12x−2

Graph y=23x−3 Graph y=−x+2

What if you have the graph, and are asked for the equation?

Once you have the equation, what does it mean in the context of the graph?

Cost of Cell Phone Plan

Once you have the equation, what does it mean in the context of the graph?

How to use the equation to solve problems:

1. Formula2. Substitution3. Algebra4. Answer

The student council sponsored a dance. A ticket costs $5 and the cost for the DJ was $300.

a. Write an equation for the profit, P dollars, on the sale of t tickets.

b. Suppose 123 tickets are sold. What is the profit?

c. Suppose the profit was $425, how many tickets were sold?

d. Could the profit be exactly $146, explain why or why not.

e. What is the break-even point?

Homework from Calendar

Section 6.5 Point – Slope Form Math 10C

The point slope form is exactly that, it is the equation for a linear relation that includes the slope and ANY POINT on that particular line. Although the equation looks a bit scary, it really is simply stating “what is the slope and what is a point”

y – y1 = m (x – x1)

When you look at the graph you want to identify 2 elements of the relation:

a.The slopei. Find 2 points and determine rise and

run through those 2 points.b.A point

i. Make sure this point is actually on the “hash-marks” of the grid so the coordinates are easily determined.

What about if you’re given the equation in the POINT SLOPE form and you are asked to graph it, what would you do?

What if you are given 2 points?

Determine the equation of the line in point slope form if it passes through the points (3, 180) and (4, 360).

1. You must determine slope.

2. Use the slope from above and 1 point (it does not matter which) and fill in the point slope form.

3. ***By simplifying the equation from above, you have now created the equation for the line.

What do you think happens if you are given a question such as this:

There will be some questions in the homework that refer to lines that are PARALLEL or PERPENDICULAR to a given line….by using this relationship, what are they actually telling us?

_ _ o _ e Knowing this, does it give you an idea of how to tackle the question?

Homework from the Calendar

Section 6.5 General Form Math 10C

Part of what you are expected to know, is how to rearrange any of the bottom equations. The key to remember is that the SAME equation for a line can be written in EACH of the three ways listed below. It is your job to:

a. Be able to identify which form is which.b. Be able to identify which form provides what information.c. Be able to convert from one form to another.

Equations of Linear Relations

Slope y-Intercept Form Point Slope Form General Form

y = mx + b (y – y1) = m (x – x1) Ax + By + C = 0

m is the slope. b is the y-Int.

m is the slope cannot determine

anything else from this form

Cannot determine slope in this form

C is the y-Int.

Let’s start with the relation y = 3x + 5 Can you change it into General Form?

What if we started with y – 1 = 35 (x + 2) Can you change it into General Form?

Because different forms provide you with different information it is important to know which form provides you with the information that you actually need.

Determine the slope of a line with the following equation:

3x – 3y – 16 = 0 5x – 2y + 12 = 0

Peanuts cost $2.00 per 100g and raisins cost $1.00 per 100g. Devon has $10.00 to purchase both those items.

a. Generate some data from this relationshipb. Graph the data.

c. Use the coordinates of two points on the line to determine slope.

d. In any method, determine the equation of the relation in general form.

e. Now use this information to answer the following:i. Will Devon be able to spend exactly $10.00 if she buys 300g fo peanuts

and 400g of raisins?

ii. Will Devon spend exactly $10.00 if she buys 400g of peanuts and 300g of raisins?

Homework from the Calendar