Algebra I Notes Unit Seven: Writing Linear Equations

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1 Sllabus Objective.6 The student will be able to write the equation of a linear function given two points, a point and the slope, table of values, or a graphical representation. Slope-Intercept Form of an Equation of a Line: = mx + b m = slope, b = -intercept Writing the Equation of a Line Given the Slope and -Intercept Ex: Write the equation of a line with a slope of 4 and a -intercept of 6. Use slope-intercept form with m = 4 and b = 6. mx b = + ( ) = 4x+ 6 or = 4x 6 Writing the Equation of a Line Given the Graph Ex: Write the equation of the line shown in the graph. The -intercept is, as seen in the graph. - - x To find the slope, use the two points shown in the graph. Starting from the point on the left, we would step down 2 and 2 right 4, so the slope is = Use slope-intercept form with m = and b=. 2 = mx+ b = x+ 2 Application Problem Ex: Xavier weighed 220 pounds and lost 2 pounds a month for 6 months. Write a linear equation to model Xavier s weight, W, over M months. Use the model to find how much Xavier weighed after 6 months. The -intercept represents Xavier s beginning weight at 0 months: b = 220 The slope (rate of change) represents Xavier s change in weight each month. (Note: Because he is losing weight, the slope will be negative.) m = 2 Equation: W = 2M To find Xavier s weight after 6 months, let M = 6. W = 2( 6) = 208 lbs. Page of McDougal Littell:..7

2 Ex: A taxi charges a flat fee of $ plus $.0 per mile. Write a linear equation that represents the cost, C, of a taxi ride for M miles. The -intercept represents the initial fee (at 0 miles): b = The slope (rate of change) represents the fee per mile: m =.0 Equation: C =.M + You Tr:. Write the equation of a line in slope-intercept form with a slope of and a -intercept of. 2. Write the equation of the line shown in the graph in slope-intercept form. A. B. - - x - - x QOD: When writing a linear model for an application problem, describe what the slope and -intercept represent. Page 2 of McDougal Littell:..7

3 Sllabus Objectives:.6 The student will be able to write the equation of a linear function given two points, a point and the slope, table of values, or a graphical representation.. The student will translate among the different forms of linear equations including slope-intercept, point-slope, and standard form..7 The student will identif parallel, perpendicular, and intersecting lines b slope. Point-Slope Form of a Linear Equation: = m( x x ) m = slope, (, ) Note: This is simpl the slope formula rewritten. x = point on the line m = x x 2 2 m x x = Cross-multipling, we have ( ) 2 2 Writing an Equation of a Line Given a Point and the Slope Ex: Write an equation of the line that passes through the point (,) and has a slope of 7 in point-slope form. x = and m = 7. Use point-slope form with (, ) (,) = ( ) ( ) m x x ( ) ( ) = 7 x = 7 x+ Ex: Write an equation of the line that passes through the point ( 2, ) and has a slope of 2 in slope-intercept form. Use point-slope form with (, ) ( 2, ) x = and m =. 2 = ( ) ( ) = ( x 2 ) + = ( x 2) m x x 2 2 To convert to slope-intercept form, solve the point-slope form equation for. + = x = x Writing an Equation of a Line Given Two Points Step One: Find the slope using the two points in the slope formula. Step Two: Use the slope and one of the points (either one) in the point-slope equation. Step Three: Write the equation in the form required for the problem. Page of McDougal Littell:..7

4 Ex: Write the equation of the line that passes through the points ( 4,7 ) and (, ) in slopeintercept form. Step One: Find the slope using the two points in the slope formula. ( ) 7 2 m = = = 4 4 Step Two: Use the slope and one of the points (either one) in the point-slope equation. We will use the point ( 4,7 ). = m( x x ) 7= 4( x 4 ) 7= 4( x 4) Step Three: Write the equation in the form required for the problem. 7= 4x 6 = 4x 9 On Your Own: Work the problem above again, using (, ) as the point in the point-slope equation. Writing an Equation of a Line Given the Graph Ex: Write the equation of the line shown in the graph in slope-intercept form. We will follow the same steps in the previous example using the points on the graph. Step One: Find the slope using the two points on the graph. Starting at the point on the left, we would step up and right 2 to get to the other point, so the slope is m = x - - Step Two: Use the slope and one of the points (either one) in the point-slope equation. We will use the point (, 0 ). = m( x x ) 0= ( x ) = ( x ) 2 2 Step Three: Write the equation in the form required for the problem. 9 = x 2 2 Slopes of Parallel Lines Activit: Graph the following lines b hand on the same coordinate plane. = x+ 6 2= ( x+ 4) 6x 2 = 2 What do ou notice about the lines? (The are parallel.) Find the slope of the lines using the equation. What do ou notice about the slopes? (The are the same.) Conclusion: Parallel lines have equal slopes. Page 4 of McDougal Littell:..7

5 Ex: Write an equation of the line parallel to the line = 2x + that passes through the point ( 4, ) in point-slope form. Step One: Determine the slope of the given line. = 2x + is in slope-intercept form with m = 2. Step Two: Determine the slope of the line parallel to the given line. Because parallel lines have equal slopes, the slope of the line parallel is also m = 2. Step Three: Write the equation of the parallel line in point-slope form using the slope found in Step Two and the point given in the problem. = ( ) ( ) = 2( x 4 ) + = 2( x 4) m x x Writing Equations of Perpendicular Lines Perpendicular Lines: two lines that intersect at a right angle. Perpendicular lines have slopes that are opposite reciprocals. Ex: Write an equation of the line perpendicular to the line 7 ( x ) the point (, 7 ) in slope-intercept form. Step One: Determine the slope of the given line. 7 ( x ) = that passes through = is in point-slope form with m =. Step Two: Determine the slope of the line perpendicular to the given line. Because perpendicular lines have slopes that are opposite reciprocals, the slope of the line perpendicular is m =. Step Three: Write the equation of the perpendicular line in point-slope form using the slope found in Step Two and the point given in the problem. = ( ) 7= ( x ) m x x Step Four: Write the equation in slope-intercept form. 7= x+ = x+ On Your Own: Graph the two lines to show the are perpendicular. You Tr: Write the equation of the line parallel to the line ( 9,6) in slope-intercept form. = x 8 that passes through the point QOD: Describe the relationship of the equations of parallel lines and perpendicular lines. Page of McDougal Littell:..7

6 Sllabus Objectives:.6 The student will be able to write the equation of a linear function given two points, a point and the slope, table of values, or a graphical representation.. The student will translate among the different forms of linear equations including slope-intercept, point-slope, and standard form. Standard Form of a Linear Equation: Ax + B = C ; where A, B, and C are real numbers and A and B are not both zero Rewriting an Equation of a Line in Standard Form Ex: Write the equation = x + in standard form with integer coefficients. 4 Step One: Wipe out fractions to create integer coefficients b multipling each term in the equation b 4. 4 = 4 x+ 4 4 = x Step Two: Rewrite the equation so that the x and terms are on one side, and the constant term is on the other side. 4 = x + 20 x+ 4 = 20 or 4 = x = x 4 x 4 = 20 Note: There is more than one wa to write an equation of a line in standard form! Writing a Linear Equation in Standard Form Given Two Points Ex: Write an equation of the line that passes through the points ( 2,4) and (, ) form. in standard Step One: Find the slope. ( ) 4 9 m = = = 2 Step Two: Use the slope and one of the points given to write the equation in point-slope form. 4= ( x ( 2) ) 4= ( x+ 2) Step Three: Rewrite the equation in standard form. 4= x 9 x+ = On Your Own: Write this equation in standard form different was. Page 6 of McDougal Littell:..7

7 Writing a Linear Equation in Standard Form Given the Graph Ex: Write the equation of the line shown in the graph in standard form. This is a vertical line with an x-intercept of 4, so the equation is x = 4. Application Problem - - x Ex: Giovanni has $20 to spend on apples and grapes at the grocer store. Apples cost $2.00 per pound and grapes cost $.2 per pound. Write a linear equation in standard form that models the different amounts of apples, a, and grapes, g, that Giovanni can bu for $20. Graph the line. Verbal Model: Price of Apples Pounds of Apples + Price of Grapes Pounds of Grapes = Total Cost Labels: Price of Apples = 2 Pounds of Apples = a Price of Grapes =.2 Pounds of Grapes = g Total Cost = Algebraic Model: 2a+.2g = 20 g 20 Graph: Let a be the horizontal axis and g be the vertical axis. The a-intercept is (let g = 0), and the g-intercept is 6 (let a = 0). An point on the line is a possible combination of apples and grapes he can bu. 20 a You Tr:. Write an equation in standard form with integer coefficients of the line that passes through the points ( 9, 2) and (, 2). 2. Write an equation in standard form of the horizontal line and the vertical line that pass through the point ( 8,9). QOD: Toda we learned that there is more than one wa to write an equation of a line in standard form. Is there more than one wa to write an equation of a line in point-slope form? In slope-intercept form? Page 7 of McDougal Littell:..7

8 Sllabus Objective:.8 The student will design, construct and analze scatter plots to make predictions..9 The student will be able to use a scatter plot to find a linear equation that approximates a set of data points. Scatter Plot: a graph of ordered pairs that represent real-life situations that is used to show relationships between the two quantities Correlation: the relationship between quantities as shown in a scatter plot There are three tpes of linear correlation: Positive Correlation Negative Correlation Relativel No Correlation Best-Fitting Line: a line that best fits the pattern of the data in a scatter plot Note: If data have positive correlation, the slope of the best-fitting line will be positive. If data have negative correlation, the slope of the best-fitting line will be negative. If data have relativel no correlation, drawing a best-fitting line is not appropriate. Writing an Equation of a Best-Fitting Line Ex: Draw a scatter plot for the age and tail length of some tadpoles. Find the equation of the best-fitting line. Determine an correlation. Age (das) Tail (mm) Step One: Draw the scatter plot b plotting the ordered pairs on a coordinate grid. We will use the age in das as the horizontal axis, a, and the tail length as the vertical axis, t. t Step Two: Sketch the best-fitting line. It should come close to all of the data points. t a a Page 8 of McDougal Littell:..7

9 Step Three: Pick two points on the best-fitting line and write the equation of the line using these two points. Two points on the line are (,4 ) and (, ). The slope of the line is 4 =. 7 =. 7 Using point-slope form, we have an equation of the line as t 4 ( a ) Rewriting in slope-intercept form, we have t 4 = a+ t = a Note: Depending on how ou drew the best-fitting line, our equation ma differ slightl. Step Four: Determine the tpe of correlation. The data have negative correlation because the slope of the best-fitting line is negative. Note: We can use the best-fitting line to make predictions about the data. Ex: In the tadpole example, predict how man das old a tadpole would be when its tail is 0 mm. Using the best-fitting line, it would take about 2 das for the length of a tadpole s tail to be 0 mm long. Graphing Calculator Activit: Scatter Plots and Linear Regression Ex: Using the tadpole data, create a scatter plot on the graphing calculator and have it compute the best-fitting line using linear regression. Step One: Enter the data from the table into the Lists L and L2. On the home screen, choose STAT then Edit. To clear data from the lists, highlight the name of the list at the top of the column and press CLEAR. Then enter the ages in L and the tail lengths in L2. Step Two: On the home screen, use the following ke strokes to calculate the best-fitting line (linear regression) and store it in Y. Step Three: Graph the scatter plot with the best-fitting line. Go to the STAT PLOT menu b pressing 2 ND Y=. Choose the options shown in the screen shot. Use the same window as our graph above. Note: The calculator s equation of the line and the one we found earlier are fairl close. Page 9 of McDougal Littell:..7

10 You Tr: Draw a scatter plot for the Women s Olmpic 0-meter times below. Find the equation of the best-fitting line. Determine an correlation. Predict the time of the women s 0-meter in the ear Year (9 ) Time (sec) QOD: Describe a real-life situation when a set of data would have a positive correlation. Page of McDougal Littell:..7

Graphs of linear equations will be perfectly straight lines. Why would we say that A and B are not both zero?

Graphs of linear equations will be perfectly straight lines. Why would we say that A and B are not both zero? College algebra Linear Functions : Definition, Horizontal and Vertical Lines, Slope, Rate of Change, Slopeintercept Form, Point-slope Form, Parallel and Perpendicular Lines, Linear Regression (sections.3