Chapter 2 Linear Relations and Functions BY: FRANKLIN KILBURN HONORS ALGEBRA 2

Summary Slide 2 – 1 Relations & Functions 2 – 1 Cont'd 2 – 2 LINEAR EQUATIONS 2 – 2 Cont'd

Summary Slide (cont.) 2 – 3 SLOPE 2 – 3 Cont'd 2 – 4 WRITING LINEAR EQUATIONS 2 – 4 Cont'd

Summary Slide (cont.) 2 – 5 Modeling Real-World Data: Using Scatter Plots 2 – 5 Cont'd 2 – 6 SPECIAL FUNCTION 2 – 6 Cont'd

Summary Slide (cont.) 2 – 7 GRAPHING INEQUALITIES 2 – 7 Cont'd Examples of Boundaries

2 – 1 Relations & Functions  Ordered pairs can be graphed on a coordinate system. The Cartesian coordinate plane is composed of the x- axis (horizontal) and the y-axis (vertical), which met at the origin (0,0) and divide the plane into four quadrants.

2 – 1 Cont'd  A relation is a set of ordered pairs.  The domain of a relation is the set of all first coordinates (x- coordinates) from all the ordered pairs, and the range is the set of all ordered coordinates from all second coordinates (y- coordinates).  The graph of a relation is the set of points in the coordinate plane corresponding to the ordered pairs in the relation.  A function is a special type of relation in which each element of the domain is paired with exactly one element of the range.  A mapping shows how each member of the domain is paired with each member of the range

2 – 1 Cont'd  A function where each element of the range is paired exactly one element of the domain is called a one-to-one function. Vertical line test: if no vertical line intersects a graph in more than one point, then the graph represents a function When an equation represents a function there are two sets of variables: The independent variable is usually x, and the values make up the domain. A dependent variable usually y, has values which depend on x.  The equations are often written in functional notation. Ex: y=2x+1 can be written as f(x)=2x+1. The symbol f(x) replaces the y and is read “f of x”.

2 – 2 LINEAR EQUATIONS A linear equation has no operations other than addition, subtraction, and multiplication of a variable by a constant. The variables may not be multiplied together or appear in a denominator. Does not contain variables with exponents other than 1. The graph is always a line.

2 – 2 Cont'd  A linear function is a function whose ordered pairs satisfy a linear equation. Any linear function can be written in the form f(x) = mx+b, where m and b are real numbers.  Any linear equation can be written in standard form – Ax+By=C – where A, B, and C are real numbers. The y-intercept is the point of the graph in which the y-coordinate crosses the y-axis.  The x-intercept is the point of the graph in which the x-coordinate crosses the x-axis.

2 – 3 SLOPE  The slope of a line is the ratio of the changes in y-coordinates to the change in x-coordinates. Slope measures how steep a line is.  A family of graphs is the group of graphs that displays one or more similar characteristics. The parent graph is the simplest of the graphs in a family

2 – 3 Cont'd The rate of change measures how much a quantity changes on average, relative to the change in another quantity. The slope of a line tells the direction in which it rises of falls: If the line rises to the right, the slope is positive. If the line is horizontal, the slope is zero. If the line falls to the right, the slope is negative. If the line is vertical, the line is undefined.

2 – 3 Cont'd In a plane, non-vertical lines with the same slope are parallel. All vertical lines are parallel. In a plane, two oblique lines are perpendicular if and only if the product of their slopes is -1.

2 – 4 WRITING LINEAR EQUATIONS  Slope – intercept form is the equation of a line in the form y=mx+b, where m is the slope and b is the y - intercept. An equation in the form y = 4/3 x - 7 is the point slope form.  The slope-intercept and point- slope forms can be said to find equations of lines that are parallel or perpendicular to given lines.

2 – 4 Cont'd The point - slope form of the equation of a line is y-y^1=m(x- x^1) where (x^1,y^1) are the coordinates of a point on the line and m is the slope of the line.

2 – 5 Modeling Real-World Data: Using Scatter Plots Data with two variables such as speed and Calories is called bivariate data. A set of bivarate date graphed as ordered pairs in a coordinate plane is called a scatter plot. A scatter plot can show whether there is a relationship between the data.

2 – 5 Cont'd A scatter plot is a set of data graphed as ordered pairs in a coordinate plane. An equation suggested by the points of a scatter plot used to predict other points is called a prediction equation. Line of fit: line that closely approximates a set of data

2 – 6 SPECIAL FUNCTION A step function is a function whose graph is a series of line segments. A greatest integer function is a step function, written as f(x)=[[x]], where f(x) is the greatest integer less than or equal to x. A constant function is a linear function in the form of f(x)=b.

Identity function: the function of 1(x)=x A piecewise function is written using two or more expressions A constant function is a linear function in the form of f(x)=b. 2 – 6 Cont'd

2 – 7 GRAPHING INEQUALITIES A linear inequality resembles a linear equation, but with an inequality symbol rather than an equal symbol. Ex: y<2x+1 is a linear inequality and y=2x+1 is the related linear equation.

2 – 7 Cont'd A boundary is a region bounded when the graph of a system of constraints is a polygonal region. Graphing absolute value inequalities is similar to graphing linear equations. The inequality symbol determines whether the boundary is solid or dashed, and you can test a point to determine which region to shade.

2 – 7 Cont'd A linear inequality resembles a linear equation, but with an inequality symbol rather than an equal symbol. Ex: y<2x+1 is a linear inequality and y=2x+1 is the related linear equation.

Examples of Boundaries Example 1 Dashed Boundary Example 2 Solid Boundary