Chapter 6: Linear Relations Section 6. Chapter 6: Linear Relations Section 6.: Slope of a Line Terminolog: Slope: The steepness of a line. Also known as the Rate of Change. Slope = Rise: The change in vertical distance as the graph etends from left to right. Rise can be either positive or negative. Run: The corresponding change in horizontal distance as the graph etends from left to right. We almost alwas read run as being positive. Determining the Slope a Line From a Graph E. Determine the Slope of Line AB A - - - - - - - B - - - NOTE: Slope should alwas be epressed as a fraction in simplest form. 7
Chapter 6: Linear Relations Section 6. E. Determine the Slope of Line CD C - - - - - - D - - - - C - - - - - - - - - D - - - - - - C - - - D - - 8
Chapter 6: Linear Relations Section 6. Slope Summar: Slope is alwas measured from left to right therefore:. When a line segment goes up and to the right, both and increase; thus both the rise and the run are positive, so the.. When a line segment goes down and to the right, decreases and increases; thus the rise is negative and the run is positive, so the. Slopes of Horizontal and Vertical Lines Horizontal Line: X Y - - - - - - - - - - Vertical Line: P - - - - - - - - - - Q 9
Chapter 6: Linear Relations Section 6. Drawing a Line Segment Given Slope and a Point To draw a line segment given a point and slope ou should do the following steps:. Plot the given point on a grid. Use the slope to determine other points that would lie on the same line segment. Connect up all points Eample : Draw a line with slope of passing through point (-,-) First Plot point (-,-) Then use the slope to plot another point Since Slope = than the Rise is positive (ie up units) and the Run is positive (ie units right) Starting at (-,-), go up units and right units and plot a new point Continue to use the slope to plot additional points. As ou recall, three points are required to confidentl plot a line. - - - - - - R un = - R ise = - (-,-) - - - - - - - - R un = - R ise = - (-,-) - - Connect up the points that ou have plotted and ou re done!!! Remember to alwas draw our line through the whole graph not just connect the dots. ALWAYS USE A STRAIGHT EDGE OR RULER TO CONNECT YOUR LINE. - - - - - - - - - - 0
Chapter 6: Linear Relations Section 6. Eample : Draw a line with slope of passing through point (-,) First Plot point (-,) Then use the slope to plot another point Since Slope = than the Rise is negative (ie down units) and the Run is positive (ie units right) Starting at (-,), go down units and right units and plot a new point Rise = - (-,) Run = - - - - - - - - - - Continue to use the slope to plot additional points. As ou recall, three points are required to confidentl plot a line. Rise = - (-,) Run = - - - - - - - - - - Connect up the points that ou have plotted and ou re done AGAIN!!! NOTE: whenever a negative slope is give, we alwas let the rise to be negative and the run to be positive. - - - - - - - - - -
Chapter 6: Linear Relations Section 6. E. Graph the line described in each situation: (a) Draw a line with slope of passing through point (-,-) - - - - - - - - - - (b) Draw a line with slope of passing through point (0,) - - - - - - - - - -
Chapter 6: Linear Relations Section 6. Determining Slope Given Two Points on a Line Slope formula can also be epressed in terms of two points on a line A(, ) and B(, ) Slope = Eample: Determine the slope of the line that passes through C(-,-) and D(,) First select one point either C or D to be point ( ie the point with coordinates (, ) Select the other point to be point ( ie the point with coordinates (, ) Plug those points into the slope formula above and solve for slope Let point C(-,-) be point (, ) and D(,) be point (, )
Chapter 6: Linear Relations Section 6. Eample: Determine the slope of the line that passes through M(-,7) and N(0,) Eample : (a) Determine the slope of the line that passes through Q(,) and R(-,) (b) Determine the slope of the line that passes through F(,-9) and G(6,-)
Chapter 6: Linear Relations Section 6. Section 6.: Slopes of Parallel and Perpendicular Lines Terminolog: Parallel Lines: Lines that do not intersect each other (ie. Never touch) Perpendicular Lines: Lines that intersect each other at a right angle (90 ) Identifing Parallel Lines When two lines have the same slope (ie. have the same rise and run) the are parallel ( ) to each other. E: Determine if AB and CD are parallel lines D - - - - - - C A - - - - B E: Determine if EF and GH are parallel lines H G F - - - - - - E - - - - 8
Chapter 6: Linear Relations Section 6. E: Line MN passes through the points M(-,) and N(,-). Line RT passes through points R(-,7) and T(7,). Are these lines parallel? E: Line PQ passes through the points P(,-) and Q(-,). Line XY passes through points X(,) and Y(-7,-6). Are these lines parallel? 9
Chapter 6: Linear Relations Section 6. Identifing Perpendicular Lines When two lines are perpendicular ( ), there slopes are negative reciprocals of each other. E. If AB has a slope of, and CD is perpendicular to it, than the slope of CD must be. E: Determine if AB and CD are perpendicular lines D - - - - - - A - - - - B C E: Determine if EF and GH are perpendicular lines H F G - - - - - - - - - E - 0
Chapter 6: Linear Relations Section 6. E: Line PQ passes through the points P(-7,) and Q(-,0). Line XY passes through points X(-,-) and Y(,). Are these lines parallel, perpendicular, or neither? Determining the slope of a line that is perpendicular to a Given Line E: Determine the slope of a line that is perpendicular to the line through E(,) and F(-,-). E: Determine the slope of a line that is perpendicular to the line through E(-,) and F(,-). Slope Summar: Given the slopes of two lines:. When the slopes of the two lines are equal to each other, we know that these lines are.. When the slopes of the two lines are negative reciprocals of each other, we know that these lines are.
Chapter 6: Linear Relations Section 6. Using Slope to Identif a Polgon E: Determine if the Parallelogram ABCD is a Rectangle A 6 B - 6 - - C 6 - D - - 6 E: Determine if the Parallelogram ABCD is a Rectangle A 6 B - 6 - - 6 - D C - - 6
Chapter 6: Linear Relations Section 6. Section 6.: Lines in Slope-Intercept Form Slope-Intercept Form: The equation of a linear function can be written in the form = m + b, where m is the slope of the line and b is the -coordinate of the -intercept. Writing Linear Equations Given its Slope and -Intercept E : The graph of a linear function has slope and -intercept of. Write an equation for this function. E : The graph of a linear function has slope 7 and -intercept of. Write an equation for this function. Graphing a Linear Function given its Equation in Slope-Intercept Form E: Graph the linear function with equation: = - - - - - - - - - -
Chapter 6: Linear Relations Section 6. E: Graph the linear function with the equation: = + Writing the Linear Equation Given its Graph Determine the equation for each of the graphs shown - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Chapter 6: Linear Relations Section 6. Using an Equation of a Linear Function to Solve a Problem E: The student council sponsered a dance. A ticket consts $ and the const for the DJ was $00. (a) Write an equation for the profit, P dollars, on the sale of t tickets. (b) Suppose people bought tickets. What was the profit? (c) Suppose the profit was $0. How man people bought tickets? (d) Could the profit be eactl $6? Justif our answer.
Chapter 6: Linear Relations Section 6. E: To join the local gm, Karin pas a start-up fee of $99, plus a monthl fee of $9. (a) Write an equation for the total Cost, C dollars, for n months at the gm. (b) Suppose Karin went to the gm for months. What was the total cost? (c) Suppose the total cost was $0. How man months did Karin use the gm? (d) Could the total cost be eactl $600? Justif our answer. 6
Chapter 6: Linear Relations Section 6. Determine the Equation of a Linear Function Given Two Points To determine the equation of a line given two points it passes through, ou must first calculate the slope, plug it into the equation, then select one of the points, plug it into our equation and solve for b (the -intercept). E. Determine the equation of the linear function that passes through the points (, ) and (8, 0). E. Determine the equation of the linear function that passes through the points (6, ) and (9, 7). E. Determine the equation of the linear function that passes through the points (6, 8) and (, ). 7
Chapter 6: Linear Relations Section 6. Section 6.: Lines in Slope-Point Form Slope-Point Form: The equation of a linear function can be written in the form = m( ), where m is the slope of the line and (, ) is a point that the given line passes through. WRITING THE EQUATION OF A LINE IN SLOPE POINT FORM What is the equation of the line that passes through the point (,-7) with slope of? What is the equation of the line that has slope of and passes though the point (-,)? 8
Chapter 6: Linear Relations Section 6. GRAPHING FROM SLOPE POINT FORMULA Sketch the graph of the line with equation: (a) = ( + ) (b) + = ( ) 0 0 8 8 6 6-0 - 8-6 - - 6 8 0 - - 0-8 - 6 - - 6 8 0 - - - - 6-6 - 8-8 - 0-0 (c) 6 = ( + 7) (d) + 6 = ( ) 0 0 8 8 6 6-0 - 8-6 - - 6 8 0 - - 0-8 - 6 - - 6 8 0 - - - - 6-6 - 8-8 - 0-0 9
Chapter 6: Linear Relations Section 6. DETERMINING THE EQUATION OF A LINE GIVEN A PARALLEL OR PERPENDICULAR LINE AND A POINT Remember the slope of two parallel lines are equal Remember the slope of two perpendicular lines are negative reciprocals of each other Eample: Write the equation of the line that is: (a) Parallel to the line = and passes through the point (,) (b) Parallel to the line = + and passes through the point (6,-) (c) Perpendicular to the line = and passes through the point (,0) (d) Perpendicular to the line = 0 and passes through the point (,-) 0
Chapter 6: Linear Relations Section 6. Determining an Equation in Slope Point Given Two Points This is slightl easier than slope-intercept form as we need onl determine the slope of the line connecting the two points and then use one of the given points to complete the equation. E. Determine the equation of the line in slope-point form that passes through the points: (a) (9, ) and (, ) (b) (,0) and (, 0) (c) (, ) and ( 7, )
Chapter 6: Linear Relations Section 6.6 Section 6.6: Lines in General Form General Form: The equation of a linear function in general form is A + B + C = 0, where A is a whole number (ie. MUST BE POSITIVE) and B and C are Integers (ie can be either positive or negative) Conversion Between Forms of a Linear Equation Slope- Intercept to General Form:. First, multipl through b the denominator of the slope.. Move everthing to the same side of the equation, making it equal to zero.. (when necessar) Multipl through b a negative if the coefficient on the -term is negative. E. Convert each to general form: (a) = (b) = + (c) = + (d) = 0 8
Chapter 6: Linear Relations Section 6.6 Slope- Point to General Form:. First, multipl through b the denominator of the slope.. Multipl the -terms b the remaining numerator using distributive propert.. Move everthing to the same side of the equation, making it equal to zero.. Combine constant terms.. (when necessar) Multipl through b a negative if the coefficient on the -term is negative. E. Convert each to general form: (a) = ( ) (b) + = ( + 7) (c) 0 = ( + ) (d) = ( 9)
Chapter 6: Linear Relations Section 6.6 General to Slope-Intercept:. Isolate the -term b moving the and constant to the other side (using inverse operations). Divide through b the coefficient on the -term.. State the new coefficient as a simplified fraction as it is the m-term (slope) E. Convert each to general form: (a) + 6 = 0 (b) 0 + 8 = 0 (d) + = 0 (d) 6 0 = 0
Chapter 6: Linear Relations Section 6.6 Graphing From General Form There are two different was to graph from general form:. Determine the and intercepts of the equation (b plugging in =0 for the - intercept and =0 for the -intercept) and drawing the line through the points as we did in chapter.. Converting the equation to slope-intercept form and graph using the slope and -intercept. Graph the line with equation: 0 (a) + = 0 8 6-0 - 8-6 - - 6 8 0 - - - 6-8 - 0 (b) 8 = 0 0 8 6-0 - 8-6 - - 6 8 0 - - - 6-8 - 0
Chapter 6: Linear Relations Section 6.6 (c) 6 8 0 = 0 0 8 6-0 - 8-6 - - 6 8 0 - - - 6-8 - 0 6