Chapter 2 Linear Relations & Functions

Chapter 2Linear Relations & Functions BY: Jered Johnson HONORS ALGEBRA 2

2 – 1Relations & Functions • 2 – 1 Cont'd • 2 – 1 Cont'd • 2 – 2 LINEAR EQUATIONS • 2 – 2 Cont'd

2 – 3 SLOPE • 2 – 3 Cont'd • 2 – 3 Cont'd • 2 – 4WRITING LINEAR EQUATIONS • 2 – 4 Cont'd

2 – 5Modeling Real-World Data: Using Scatter Plots • 2 – 5 Cont'd • 2 – 6SPECIAL FUNCTION • 2 – 6 Cont'd

2 – 7GRAPHING INEQUALITIES • 2 – 7 Cont'd • 2 – 7 Cont'd • Examples of Boundaries

2 – 1 • 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 – 4WRITING 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 – 5Modeling 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 – 6SPECIAL 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.

2 – 6 Cont'd • 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 – 7GRAPHING 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.

Examples of Boundaries • Example 1 Dashed Boundary • Example 2 Solid Boundary

Chapter 2 Linear Relations & Functions