The Coordinate Plane x y 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 Origin (0,0)
Quadrants x y I II III IV 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. (+,+) (–,+) (–, –) (+, –)
Coordinates 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. x y The first coordinate tells how many units to move left or right; the second coordinate tells how many units to move up or down. (2, 3) origin move right 2 units move up 3 units (0, 0) (2, 3) To graph an ordered pair means to draw a dot at the point on the coordinate plane that corresponds to the ordered pair. x-coordinate move right or left y-coordinate move up or down
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 its 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
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 – 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.
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 >.