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Interactive Study Guide for Students Chapter 2: Linear Relations and Functions Section 1: Relations and Functions Graph Relations Examples _______________: can be used to graph a relation of a series of two numbers. State the domain and range of the relation shown in the graph. Is the relation a function and why? Cartesian coordinate plane: composed of the x-axis and the y-axis which meet at the origin (0,0) and divide the plane into four ____________. 1. Relation: set of ordered pairs. The __________ is the set of all the first coordinates (x-coordinates) from the ordered pairs, and the ______ is the set of all second coordinates (y-coor.) __________: special type of relation where each element of the domain is paired with exactly one element of the range. When the relation has each element of the range being paired with exactly one element of the range, it is called a one-to-one ____________ one-to-one function(not one-to-one) 2. not a function The ____________ test can also be used to determine if it is a function (f it intersects only one time, it is a function). Graph the relation, find the domain and range, and state if it is a function and why? 3. y = 2x + 1 4. x = y2 -2 Equations of Functions and Relations In a function, the ___________variable is the x or the domain. The ___________ variable depends of the domain. ______________ notation: Instead of y=2x+1, writef(x)=2x + 1 Interactive Study Guide for Students 5. Given f(x)=x2-2, Find f(3). Chapter 2: Linear Relations and Functions Section 2: Linear Equations Identify Linear Equations and Functions Examples ______equations: have no operations other than addition, subtraction and multiplication of a variable by a constant. Examples Non-examples 5x – 3y = 7 7a + 4b2 = -8 y=½a y= x=5 State whether each function is a linear function. Explain. 1. f(x) = 10 – 5x 2. g(x) = x4 -5 3. h(x,y) = 2xy + 1 x5 Write each equation in Standard Form. Identify A, B, and C. 2 x= x A linear __________ is a function that satisfies the linear equation, or can be written f(x) = mx + b 4. y = -2x + 3 3 5 Standard Form 5. - x = 3y – 2 Ax + By = C where A 0, and A & B are not both zero 6. 3x – 6y -9 = 0 The y-coordinate of the point at which a graph crosses the y-axis is called the y-________. Likewise, the x-coordinate of the point at which it crosses the x-axis is the x-_________. Find the x-intercept and the yintercept of the graph of 3x – 4y + 12 = 0. Then graph the equation. Interactive Study Guide for Students Chapter 2: Linear Relations and Functions Section 3: Slope Slope Examples The _____________ of a line is the ratio of the change in ycoordinates to the corresponding change in x-coordinates. It measures how steep a line is, and is often referred to as rate of _____________. The slope m of the line passing through (x1, y1) and (x2, y2) is given by: 1. Find the slope of the line that passes through (-1, 4) and (1, -2). Graph. M = ___________ , where x1 ≠ x2 2. Graph the line passing through (-4, -3) with a slope of 2 . 3 Parallel and Perpendicular Lines A _______________of graphs is a group of graphs with one or more similar characteristics. The _________graph is the simplest one. 3. Graph the line through (-1, 3) that is parallel to the line with equation x + 4y = -4 __________ lines: non-vertical lines with the same slope. ______________ lines: two oblique lines with the product of their slopes equal to -1. m1m2= -1 or m1 = - 1 m Interactive Study Guide for Students Chapter 2: Linear Relations and Functions Section 4: Writing Linear Functions Forms of Equations Examples _________-___________form: y = mx + b where m is the slope and b is the y-intercept. If you are given the coordinates of two points on a line, you can use the ________-_________ form to find an equation of the line that passes through them. 1. Write an equation in slopeintercept form for the line that has a slope of - 3 and 2 passes through (-4, 1). The point-slope form of the equation of a line is: y – y1 = m(x – x1) where (x1, y1)are the coordinates of a point on the line and m is the slope of the line. 2. What is the equation of the line through (-1, 4) and (-4, 5)? 3. As a salesperson, Eric is paid a daily salary plus commission. When his sales are $1000, he makes $100. When his sales are $1400, he makes $120. Write a linear equation to model this situation. 4. Write and equation for the line that passes through (-4, 3) and is perpendicular to the line whose equation is y=-4x-1 Parallel and Perpendicular Lines The slope-intercept and point-intercept forms can be used to find equations of lines that are parallel or perpendicular to given lines. Interactive Study Guide for Students Chapter 2: Linear Relations and Functions Scatter Plots Section 5: Using Scatter Plots Examples To model data with a function, it is helpful to graph the data. A set of data graphed as ordered pairs in a coordinate plane is called a _________ _________. 1. The table shows the median selling price of new, privately owned, one-family homes. Make a scatter plot of the data Yr 90 92 94 96 98 2000 $$ 122.9 121.5 130.0 140.0 152.5 169.0 Prediction Equations When you find a line that closely approximates a set of data, you are finding a __________ of ____. An equation of such a line is often called a ______________ ____________ because it can be used to predict one of the variables given the other variable. Interactive Study Guide for Students Chapter 2: Linear Relations and Functions Step Functions 2. The table show households that have a TV (in millions) Draw a scatter plot, use two ordered pairs to write a prediction equation, predict the missing value. Yr 90 92 94 96 98 00 Houses 55 57 59 65 67 ? Section 6: Special Functions Examples A step function is not linear. It consists of line segments or rays. The _________ ___________function, written f(x)= [[x]], is an example of a step function. The symbol ________ means the greatest integer less than or equal to x. Constant Functions In the function f(x)=b, when m=0, the value of the function is just f(x)=b for every x value. So f(x)=b is called the ______________ function. The Identity Function Since the function f(x) = x does not change the input value, it is called the _______________function. The Absolute Value Function A type of piecewise function is the Absolute Value Function. It is called a ____________ function because it can be written using two or more expressions Interactive Study Guide for Students Chapter 2: Linear Relations and Functions Section 6: Special Functions Step Functions Examples Parent Function: f(x) = [[x]] 1. f(x) = [[x + 3]] x y D: Domain: R: Range: 2. f(x) = 3[[x]] Piecewise Functions f(x) = x – 4 if x < 2 1 if x > 2 D: x y R: 3. f(x) = x+3 if x < -1 2x if x> -1 Domain: D: Range: R: The Absolute Value Function Parent Function: f(x) = |x| x y 4. f(x) = |x-2| + 1 Domain: D: Range: R: Interactive Study Guide for Students Chapter 2: Linear Relations and Functions Section 7: Graphing Inequalities Graph Linear Inequalities The graph of a linear inequality resembles a linear equation, only it acts as a boundary of the region. Examples 1. 2x + 3y > 6 Step 1: Determine if line is solid or dashed. Step 2: Check a point not on the line. Step 3: If true; shade region of point. If false: shade other side. 2. y < 2x - 1 Graph Absolute Value Inequalities 3. y < |-2x| Graph similar to linear inequalities Step 1: Determine if line is solid or dashed Step 2: Check a point not on the line. Step 3: If true; shade region of point. If false: shade other side. x y 4. y > |x| - 2