Geometry Unit 2 Lesson 4, 5 and 6 Notes

Lesson 4 Graphs of Algebraic Sentences

Linear equation

The well-known equation y = mx + b is called a linear equation because the graph of all possible solutions for any linear equation is a straight line.

The “m” variable in the equation represents the slope of the line and the “b” variable is the location where the line crosses the y-axis. 

Linear inequality

To graph a linear inequality use the same steps as in graphing a linear equation. If the inequality symbol is less than or greater than use a dashed line when drawing a line. If the inequality symbol is less than or equal to or greater than or equal to use a solid line. 

The big difference in graphing equations versus inequalities is the range of possible answers. 

Graphing Linear Equations

You have learned that the equation of a line is "y equals mx plus b," where m is the slope of the line and b is the y-intercept. There are two types of lines where this equation is modified- horizontal and vertical lines.

The example "x = 3." This is definitely not in the traditional "y equals mx plus b"

form. The y variable has dropped out the equation, so it has no bearing on the problem. The condition for this graph is that for every ordered pair, the x-coordinate must be 3. To show all possible points for this equation, we draw a vertical line through x equals 3. 

The same holds true for an equation that reads "y = 4," except now, all the y-coordinates must be 4. This creates a horizontal line through "y = 4."

Graphing inequalities

Model 1: Graph {(x, y): y   2}.

This is read "find all x and y values such that y is less than or equal to 2".

Solution: First, graph the line y = 2 (we will deal with the fact that the equation is actually an inequality later.) Now, since the equation actually asks for all the ordered pairs that have a y value less than or equal to 2, we must shade the graph to represent this. We show the line as a solid line because y = 2 will meet the requirements (if the equation was y   2) we would show the line as a dotted line to reflect that y = 2 would not satisfy the equation).

All points on the line and in the shaded region have a y-coordinate that is less than or equal to 2. No restrictions are placed on the x-coordinate.

Model 2: Graph {(x, y): -5   x   1}.

This is read "Find all the ordered pairs (x, y) such that x is greater than -5 and less than or equal to 1."

Solution: First, draw the two lines x = -5 and x = 1. The line x = -5 will be a dotted line because x = -5 does not satisfy the inequality. The x = 1 line will be a solid line because x = 1 does satisfy the inequality.

The final step is to determine which portion of the graph is shaded. The easiest way to accomplish this is to pick a point and see if it satisfies the inequality. For example, pick the point (0, 0) which lies between the two lines. Does (0, 0) satisfy the inequality?

Is -5   0   1 a true statement? Yes, it is. Therefore, the point (0, 0) satisfies the inequality and we would shade the space between the two lines.

All points on the solid line and in the shaded region meet the condition on x. The points on the broken line do not meet those conditions for x.

Practice:

1. Graph {(x, y): x = 3}   {(x, y): y   2}.

This is read: "Find the ordered pairs (x, y) that represent the union of x = 3 and y   2.

2. Graph {(x, y): x = 3}   {(x, y) y   2}.

This graph will represent the intersection of the two lines.

3. (Check with Odysseyware )

Lesson 5 Distance Formula

To compute any nonvertical, nonhorizontal length, you could draw a right triangle and use the Pythagorean Theorem. In order to accomplish this, you would have to square the difference of x values and add this to the square of the difference of the y-values and then take the square-root of that difference. 

Since triangle A B C is a right triangle, you can use the Pythagorean Theorem to calculate the length of segment A B. 

Distance Formula

To find the distance between two points that have the same y-coordinate, we take the absolute value of the difference in their x-coordinates.

P1P2 = |6 - 2| = |4| = 4

P3 P4 =|-5 - (-2)| = |-3| = 3In general, when both points are on the same horizontal line, P1P2 = |x 1 - x 2|.

To find the distance between two points that have the same x-coordinate, we take the absolute value of the difference in their y-coordinates.

P1P2 = |2 - 5| = |-3| = 3P3P4 = |2 - (-5)| = |2 + 5| = |7| = 7In general, when both points are on the same vertical line, P1P2 = |y 1 - y 2|.

A third possibility is that the two points are not on the same horizontal line or the same vertical line. We then find the distance with the help of the Pythagorean Theorem.

Complete a right triangle with P1P2 its hypotenuse and P1Q, P2Q its two legs. The coordinate of point Q will be (x 1, y 2). It has the same x-coordinate as P1 and the same y-coordinate as P2. Next, we find the distances P1Q and P2Q. P1Q = |y 1 - y 2|, P2Q = |x 1 - x 2 |. To find P1P2, we apply the Pythagorean Theorem to the right triangle:

(P1P2)2 = (P2Q)2 + (P1Q)2

(P1P2)2 = (x 1 - x 2)2 + ( y 1 - y 2)2

THEOREM 9-1 The distance between two points on the coordinate plane is

where ( x 1, y l) and ( x 2, y 2) are the coordinates of the two points.

Using the Distance Formula

If point D is at (7, 1), point E is at (9, negative 5), and point F is at (6, negative 4), can we find the length of the median from F to segment D E? First, we need to recall that a median connects a vertex of a triangle with the midpoint of the opposite side.So, we need to calculate the midpoint of side D E, and then draw in the median on our diagram.

The midpoint is found by calculating the average of the x-values and the average of the y-values for the endpoints of the segment in which you wish to find the midpoint.

The calculation produces (8, negative 2) as the midpoint of segment D E.

Now that we know the location of point F and point M, we can use the distance formula to calculate the length of segment F M.

F M equals the square root of the change in the x-coordinates squared, plus the change

in the y-coordinates squared.

By drawing in the median, can we tell if the two smaller triangles are congruent?

Lesson 6 Perimeter and Area

distance formula distance between two points: 

perimeter the distance around the border of a two-dimensional shape