Algebra ( Period 5)

Chapter 3 Review ( After Thanksgiving)

You know 2 ways to graph:

Intercepts and 3 random points

You can find the x intercepts = solutions = roots = zeros by graphing or algebraically ( setting y = 0 and solving for x)

The slope of a linear equation is also known as the rate of change

On a line, the slop, or rate of change is a constant amount between any two points.  To find the slope ( the rate of change) either count the rise over the run between any two points OR

use the slope formula which find the change in y/ change in x

y₂ – y₁

x₂ – x₁

Special Lines reviewed with their Slopes
HORIZONTAL LINES have only a y intercept (unless it’s the line y = 0 and then that is the x axis The equation of a horizontal line is y = b where b is a constant

Notice that there is no x in the equation. y = 4 is a horizontal line parallel to the x axis 4 steps above the x axis. The y value is always 4 What is the x value? ALL real numbers!

The constant rate of change  or SLOPE = 0

VERTICAL LINES (which are NOT functions) have only an x intercept (unless it’s the line x = 0 and then it’s the y axis) The equation of a vertical line is x = a, where a is a constant. Notice that there is no y in the equation. For example x = 4 is a vertical line parallel  to the y axis where the x value is always 4. What is the y value?  All real numbers. The constant rate of change or slope is UNDEFINED!!!

If a graph or equation goes through the origin ( 0,0)  it is proportional and the ratio of any y value to its x value is a constant ( which turns out to the unit rate or constant rate of change or the slope of the line) The x and y intercepts are therefore both 0. When the linear relationship is proportional, we say that it is  DIRECT VARIATION

Now the constant rate of change, the slope, the unit rate, is called the CONSTANTOF VARIATION OR the CONSTANT OF PROPORTIONALITY. (Remember it is just new vocab!) WE also say y varies directly with x. The slope is now represented by the letter k instead of m

Finding the Equation of a Line that is Proportional

Find k( the slope) by counting the rise/ run on the graph

Write the equation using the format y = kx

Notice that you always pick the origin as the point to count rise/run the slope (k) is always just y/x

Determining whether a Table of Values is Direct Variation

If given a table of values you can determine I the relationship is direct variation by dividing 3 y’s by their x values and making sure that you get the same value. If you do, it is proportional—it goes through the origin, and the slope of y/x is the unit rate—now called the constant of variation.

Finding Additional Values for the Direct Variation once you have the Equation

Once you have the equation y = kx you can find infinite additional values ( points) that will work. remember the babysitting example?  we have y = $7.50x or y = 7.5x

If she babysat for 20 hours how much did she earn? x = 20 so y = 7.5(20) y = 150

So she earned $ $150.

Finding the Equation if you know one point and then Finding Additional Values

y varies directly with x. Write an equation for the direct variation. Then find each value. If y = 8 x = 3, find y when x = 45

Solution since it is  direct variation, the slope x = y/x
K = 8/3

Equation is y = (8/3)x

Now find y when x = 45

y = (8/3)(45)

y = 120

 go through the origin (0,0) and nonproportional do NOT

Nonproportional have a y-intercept other than 0

Proportional relationships: YOU can take any point and divide the y/x and it will equal the same value as diving any other y/x. This value  is the slope—which is the constant rate of change  VS Nonproportional relationships when you divide the y/x of a point it will NOT equal the y/x of another point. This value is NOT the slope and is NOT the CONSTANT RATE OF CHANGE

TO find the equation for anonproportional relationship
This isn’t as easy as f(x) = kx because it does not go through (0,0)

You will need to find the y intercept (0,y)

Say you find the rate of change or slope is 3 for the following 2 points

(2, 12) and (4, 18)

You can graph these two points and count the slope down to the y intercept

You can find a missing number that will make the equation work

y = 3x + ?  will make ( 2, 12) work in the equation

Plug in( 2, 12) to find b

12 = 3(2) + b

12 = 6 +b

 b = 6

You can try it with other points as well…

 It still works b = 6

So the non-proportional equation is y = 3x + 6

6 is the y intercept on the graph