Presentation on theme: "Chapter 11: Graphing Lines Regular Math. Section 11.1: Graphing Linear Equations A linear equation is an equation whose solutions fall on a line on."— Presentation transcript:
Section 11.1: Graphing Linear Equations A linear equation is an equation whose solutions fall on a line on the coordinate grid. A linear equation’s graph will always be a straight line.
5 Steps to Graph any Equation 1.Choose a value for x. 2.Substitute the x-value into the equation, and find the corresponding y-values. 3.Form an ordered pair with the x-value and y-value. 4.Graph the ordered pair. 5.Repeat the process until you have at least 3 points. Remember – One point must be a negative point.
Graphing Equations Graph each equation and tell whether it is linear. y = 2x – 3 y = x squared y = 2/3 x y = -3
Try these on your own… Graph each equation and tell whether is it linear. y = 3x -1 Linear y = x cubed Not Linear y = -3/4 x Linear y = 2 Linear
Sports Application In bowling, the equation h = 160 – 0.8s represents the handicap (h) calculated for a bowler with average score (s). How much will the handicap be for each bowler listed in the table? Draw a graph that represents the relationship between the average score and the handicap. Bowler Average Score Sandi145 Dominic125 Leo160 Sheila140 Tawana175
Try this one on your own… A lift on a ski slope rises according to the equation a = 130t + 6250, where a is the altitude in feet and t is the minutes that a skier has been on the life. Five friends are on the lift. What is the altitude of each person if they have been on the ski lift for the times listed in the table? Draw a graph that represents the relationships between the time on the lift and the altitude. Skier Time of Lift Anna 4 minutes Tracy 3 minutes Kwani 2 minutes Tony 1.5 minutes George 1 minute
Skier Time on Lift Altitude Anna 4 minutes 6770 ft Tracy 3 minutes 6640 ft Kwani 2 minutes 6510 ft Tony 1.5 minutes 6445 ft George 1 minute 6380 ft
Finding Slope, Given To Points Find the slope of the line that passes through (2,5) and (8,1). Try this one on your own… Find the slope of the line that passes through (-2, - 3) and (4, 6).
Finding Slope from a Graph Use the graph of the line to determine its slope.
Try this one on your own… Use the graph of the line to determine its slope.
Parallel and Perpendicular Slopes Parallel Lines have the same slope. Perpendicular Lines have complete opposite slopes.
Identifying Parallel and Perpendicular Lines by Slope Identifying Parallel and Perpendicular Lines by Slope Line 1: (1,9) and (-1,5) Line 2: (-3, -5) and (4,9) Line 1: (-10, 0) and (20,6) Line 2: (-1, 4) and (2, - 11)
Graphing a Line Using a Point and the Slope Graph the line passing through (1,1) with slope -1/3. Graph the line passing through (3,1) with slope 2.
Section 11.3: Using Slopes and Intercepts The x-intercept of a line is the value of x where the line crosses the x-axis. (y = 0) The y-intercept of a line is the value of y where the line crosses the y-axis. (x = 0)
Finding x-intercepts and y-intercepts to Graph Linear Equations Find the x-intercept and y-intercept of the line 2x + 3y = 6. Use the intercepts to graph the equations. Step One: Solve for y. Step Two: Find the x – intercept and the y – intercept. Step Three: Graph.
Try this one on your own… Find the x-intercept and y-intercept of the line 4x – 3y = 12. Use the intercepts to graph the equation.
Slope – Intercept Form Slope – Intercept Form : y = mx + b m = slope b = y-intercept Notice that y is all by itself on one side and everything else is on the other.
Using Slope-Intercept Form to Find Slope and y-intercept Write each equation in slope-intercept form, and then find the slope and y- intercept. y = x 7x = 3y 2x + 5y = 8
Try these on your own… Write each equation in slope-intercept form, and then find the slope and the y- intercept. 2x + y = 3 5y = 3x
Entertainment Application An arcade deducts 3.5 points from your 50-point game card for each Skittle-ball game you play. The linear equation y = -3.5x + 50 represents the number of points (y) on your card after (x) games. Graph the equation using the slope and y-intercept.
Try this one on your own… A video club charges $8 to join, and $1.25 for each DVD that is rented. The linear equation y = 1.25x + 8 represents the amount of money (y) spent after renting (x) DVDs. Graph the equation using the slope and y – intercept.
Writing Slope-Intercept Form Write the equation of the line that passes through (-3,1) and (2, -1) in slope- intercept form. Try this one on your own… Write the equation of the line that passes through (3, -4) and (-1,4) in slope-intercept form.
Section 11.4: Point-Slope Form The point-slope form of an equation of a line with slope (m) passing through (x1,y1) is y – y1 = m (x – x1).
Use Point-Slope Form to Identify Information About a Line Use the point-slope form of each equation to identify a point the line passes through and the slope of the line. y – 9 = -2/3 (x - 21) m = -2/3 Point = (21, 9) y – 3 = 4 (x + 7) m = 4 Point = (-7, 3)
Try these on your own… Use the point-slope form of each equation to identify a point the line passes through and the slope of the line. Y – 7 = 3 (x – 4) m = 3 Point = (4,7) Y – 1 = 1/3 ( x + 6) m = 1/3 Point = (-6, 1)
Writing the Point-Slope Form of an Equation Write the point-slope form of the equation with the given slope that passes through the indicated point. the line with slope -2 passing through (4,1) the line with slope 7 passing through (-1,3)
Try these on your own… Write the point-slope form of the equation with the given slope that passes through the indicated point. the line with slope 4 passing through (5, -2) y + 2 = 4 (x – 5) the line with slope -5 passing through (-3, 7) y – 7 = -5 (x + 3)
Medical Application Suppose that laser eye surgery is modeled on a coordinate grid. The laser is positioned at the y-intercept so that the light shifts down 1 mm for each 40 mm it shifts to the right. The light reaches the center of the cornea of the eye at (125,0). Write the equation of the light beam in point-slope form, and find the height of the laser.
Try this one on your own… A roller coaster starts by ascending 20 feet for every 30 feet in moves forward. The coaster starts at a point 18 feet above the ground. Write the equation of the line that the roller coaster travels along in point-slope form, and use it to determine the height of the coaster after traveling 150 feet forward. Assume that the roller coaster travels in a straight ling for the first 150 feet.
Section 11.5: Direct Variation For direct variation, two variable quantities are related proportionally by a constant positive ratio. The ratio is called constant of proportionality. Equation: y = kx k = constant
Determining Whether a Data Set Varies Directly Determine whether the data set shows direct variation. Shoe Sizes… US Size7891011 European Size 3941434445
Determine whether the data set shows direct variation. Distance Sound Travels at 20 degrees Celcius (m) Time (s)01234 Distance (m) 035070010501400
Try these on your own… Determine whether the data set shows direct variation. Adam’s Growth Chart Distance Traveled by Train Time (Min)10203040 Distance (mi) 255075100 Age (mo)36912 Length (in.)22242527
Finding Equations of Direct Variation Find each equation of direct variation, given that y varies directly with x. y is 52 when x is 4 x is 10 when y is 15 y is 15 when x is 2
Try these on your own… Find each equation of direct variation, given that y varies directly with x. y is 54 when x is 6 x is 12 when y is 15 y is 8 when x is 5
Story Problem… Mrs. Perez has $4000 in a CD and $4000 in a money market account. The amount of interest she has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either data set and time. If so, find the equation of direct variation. Time (mo)Interest in CD ($) Interest in Money Market ($) 000 11719 23437 35155 46873
Section 11.6: Graphing Inequalities in Two Variables When the equality symbol is replaced in a linear equation by an inequality symbol, the statement is a linear inequality. A boundary line is the set of points where the two sides of a two-variable linear inequality are equal.
Graphing Inequalities Graph each inequality.
Try these on your own… Graph each inequality.
Science Application… Solar powered rovers landing on Mars in 2004 will have a range of up to 330 feet per Martian day. Graph the relationship between the distance a rover can travel and the number of Martian days. Can a rover travel 3000 feet in 8 days?
Try this one on your own… A successful screenwriter can write no more than seven and a half pages of dialogue each day. Graph the relationship between the number of pages the writer can write and the number of days. At this rate, would the writer be able to write a 200 page screenplay in 30 days?
Section 11.7: Lines of Best Fit To estimate the equation of a line of best fit: Find the mean of the x-coordinates and y- coordinates. Create a new point. Draw a line through the new point that appears to fit the data the best. Estimate the coordinates of another point on the line. Find the equation of the line.
Finding a Line of Best Fit XY 24 48 57 13 34 88 65 79
Sports Application Find a line of best fit for the women’s 3000-meter speed skating. Use the equation of the line to predict the winning time in 2006. Let 1960 represent year 0. YearWinning Time (minutes) 19645.25 19684.94 19724.87 19764.75 19804.54 19844.41 19884.20 19924.33 19944.29 19984.12 20023.96
Try this one on your own… Find a line of best fit for the Main Street Elementary annual softball toss. Use the equation of the line to predict the winning distance in 2006. Let x = 0 represent the year 1990. YearDistance (ft) 199098 1992101 1994103 1997106 2002107