Review Chapter 4 Sections 1-6

The Coordinate Plane 4-1

Vocabulary Axes Origin Coordinate plane Y-axis X-axes X-coordinate Y-coordinate Quadrant Graph

The Coordinate Plane Axes – two perpendicular number lines. Origin – where the axes intersect at their zero points. X-axes – The horizontal number line. Y-axis – The vertical number line. Coordinate plane – the plane containing the x and y axes. 1 2 3 4 5 -1 -2 -3 -4 -5 y x 1 2 3 4 5 -1 -2 -3 -4 -5 Origin (0,0)

Quadrants 1 2 3 4 5 -1 -2 -3 -4 -5 y Quadrants – the x-axis and y-axis separate the coordinate plane into four regions. Notice which quadrants contain positive and negative x and y coordinates. II I (–,+) (+,+) x 1 2 3 4 5 -1 -2 -3 -4 -5 III IV (–, –) (+, –)

Coordinates (2, 3) (2, 3) (0, 0) origin move right 2 units 1 2 3 4 5 -1 -2 -3 -4 -5 y To plot an ordered pair, begin at the origin, the point (0, 0), which is the intersection of the x-axis and the y-axis. (2, 3) (0, 0) x The first coordinate tells how many units to move left or right; the second coordinate tells how many units to move up or down. 1 2 3 4 5 -1 -2 -3 -4 -5 origin move right 2 units (2, 3) move up 3 units x-coordinate move right or left y-coordinate move up or down To graph an ordered pair means to draw a dot at the point on the coordinate plane that corresponds to the ordered pair.

Transformations on the Coordinate Plane 4-2

Vocabulary Transformation – movements of geometric figures Preimage – the position of the figure before the transformation Image – the position of the figure after the transformation. Reflection – a figure is flipped over a line (like holding a mirror on it’s edge against something) Translation – a figure is slid in any direction (like moving a checker on a checkerboard) Dilation – a figure is enlarged or reduced. Rotation – a figure is turned about a point.

Types of Transformations

Reflection and Translation

Dilation and Rotation

Relations 4-3

Vocabulary Mapping – a relation represented by a set of ordered pairs. Inverse – obtained by switching the coordinates in each ordered pair. (a,b) becomes (b,a) Relation – a set of ordered pairs

Mapping, Graphing, and Tables

Mapping the Inverse

Equations as Relations 4.4

Vocabulary Equation in two variables – an equation that has two variables Solution – in the context of an equation with two variables, an ordered pair that results in a true statement when substituted into the equation.

Different Ways to Solve Solving using a replacement set – a variation of guess and check. You start with an equation and several ordered pairs. You plug each ordered pair into the equation to determine which ones are solutions. Solving Using a Given Domain – Start with an equation and a set of numbers for one variable only. You then substitute each number in for the variable it replaces, and solve for the unknown variable. This gives you a set of ordered pairs that are solutions.

Dependent Variables When you solve an equation for one variable, the variable you solve for becomes a “Dependent Variable”. It depends on the values of the other variable. Dependent Variable Independent Variable The values of “y” depend on what the value of “x” is.

Graphing Linear Equations 4.5

Vocabulary Linear equation – the equation of a line Standard form – Ax + By = C where A, B, and C are integers whose greatest common factor is 1, A is greater than or equal to 0, and A and B are both not zero. X-intercept – The X coordinate of the point at which the line crosses the x-axis (Y is equal to 0) Y-intercept – the Y coordinate of the point at which the line crosses the y-axis (X is equal to 0)

Methods of Graphing Make a table – Use the Intercepts – Solve the equation for y. Pick at least 3 values for x and solve the equation for the 3 values of y that make the equation true. Graph the resulting x and y (ordered pair) on a coordinate plane. Draw a line that includes all points. Use the Intercepts – Make X equal to zero. Solve for Y. Make Y equal to zero. Solve for X. Graph the two coordinate pairs: (0,Y) and (X,0) Draw a line that includes both points.

Functions 4.6

Vocabulary Function – a relation in which each element of the domain is paired with exactly one element of the range (for each value of x there is a value for y, but each value of y cannot have more than one value of x) Vertical line test – if no vertical line can be drawn so that it intersects the graph in more than one place, the graph is a function Function notation – f(x) replaces y in the equation.

Vertical Line Test

Function Notation f(5) =3(5)-8 =15-8 =7

Other Functions and Notations Non-Linear Functions – Functions that do not result in a line when plotted. Alternative Function Notation – another way of stating f(x) is <<x>>.