Math 00 Lesson -4 (Textbook.7 and.8) Different equations of Lines. slope intercept form = mx + b. standard form Ax + B = C m x x 3. point slope form ( ) Which one we use depends on what information is given to us in the problem. x is alone. x The constant is alone. 4 ( x ) Neither x nor is alone.
Slope Intercept form of an equation of a line: the coordinates of all points (x, ) on a line whose average rate of change is m and whose -intercept is b. mx b f ( x) mx b 9) If ou substitute x = 0 into the equation above, what is the corresponding value of? 0) What is the slope and -intercept for the lines given b the following equations? (Give -intercepts as x- pairs) f ( x) x s 0.75t 5 q r 5 x 6 3
(-, -4) (0, ) (, 8) f ( x) 3x The constant is the -intercept 8 - -4 0
hat is the equation of the line? x x 3 - - -
What is the equation of the line? (-3,) (-,) (,) (3,) x or? Which value (x or ) is alwas?
3- mile Time (minutes) Time in Program (weeks) Horizontal intercept (x-intercept), the point where the graph crosses the input axis. It alwas has (what value?) as the -value of the point. Numericall it is alwas: (a, 0) or (x, 0) In function notation it is alwas: f ( x) 0
Example: Find the x-intercept for each of the following equations. One step rewrite show our work!!! f s ( x) x 5 0 = -x + 5 +x +x x = 5 x 5 0.75t q r 5 x 6 3
Graph an equation of a line: 3 x 3 ) Calculate the intercepts: -intercept: f(0) = (0) 3 3 (0, 0 x x-intercept: f(x) = 0 3-3 -3 3 x 3) *(-) *(-) 3 4 5 6 6 x (6, 0)
Graph an equation of a line: 3 x 3 ) = mx + b -intercept: f(0) = b slope: (0, 3) m 3 4 5 6 From -intercept, move down, right. (, )
Point Slope Equation of a Line Slope is given b: Multipl b x ) x ) ( x m( x m m( x x ) *( x x ) ( x x ) x x x m( x ) If we know one x- pair (for example: (, 3)) and the slope (for example: m = ) this becomes: 3 ( x ) We can drop the subscripts: 3 ( x ) Convert to slope intercept form: 3 x +3 +3 x
Point Slope Equation of a Line x m( x ) Convert the following equations into slope intercept form (one step rewrite). 4 3( x ) 4 3x 6 +4 +4 3x 3 6 ( x 4) 3 6 x 6-6 -6 3 x
Point Slope Form of a Linear Equation What is the equation of a line that passes through the point (3, 4) and has a slope of -? Step : write the general form of the equation. m x x or m( x x ) Step : substitute numbers into the equation. ( x, ) 4 ( x 3) (3,4) m = - x 0 Step 3: Solve for (slope/int form).
Time (min) 0 3 4 Height (ft) 36,000 3,000 8,000 4,000 0,000 What is happening? Write an Equation what is the slope? m rise run 4000 ft min What is the -intercept? (0, b) (0, 36000) write equation: = mx + b = -4000x + 36000
Time (min) 8:03 AM 8:04 AM 8:05 AM 8:06 AM 8:07 AM Height (ft) 36,000 3,800 9,600 6,400 3,00 Notice how this time doesn t start at zero. To write an equation, ou need a -intercept. It is often easier to change the time scale to read time since some reference point. Time (min) 0 3 4 (since 8:03 AM) Height (ft) 36,000 3,800 9,600 6,400 3,00
Year 990 99 994 996 998 Imports (Billions $) 5 55 58 6 64 Write an Equation what is the slope? m rise run 3 What is the -intercept? (0, b) Change scale for ears to ears since 990) (0, b) Years since 990 0 4 6 8 Imports (Billions $) 5 55 58 6 64 write equation: =.5x + 5
Finding an equation of a line that passes through to given points. (3, 4) and (7, 0) Which form will work the easiest? slope-intercept? standard form? point-slope?
Equations of Parallel Lines How do the slopes of parallel lines compare? Parallel same slope Find the equation of a line that is parallel to the line = x + and passes through the point (4, -) What equation form will ou use? ( x, ) (4, Slope = ) m( x x ) ( ) ( x 4) x 8 x 9
Perpendicular Lines How do the slopes of perpendicular lines compare? Slopes are reciprocals of each other. m =? m undefined 0 m =? m = 0
Perpendicular Lines Positive or negative slope? Positive or negative slope?
Perpendicular Lines: The slopes are reciprocals of each other. The slopes have opposite signs of each other. m = -3 What is the slope of a line that is perpendicular? Find the slope intercept form of a line that is perpendicular to the line: = x 6 and passes through the point (7, )
You decide to bu a new Honda Civic, but ou are concerned about the value of the car depreciating over time. You search the Internet and obtain the following information. Suggested Retail Price: $0,905 Depreciation per ear: $750 (assume constant) ) What does this mean? ) Complete the table. V is the value of the car after n ears of ownership n (ears) 0 3 5 8 V, ($) 0,905 9,55 7405 5,655,55 6905 3) Is the value of the car a function of ears of ownership? Explain wh or wh not.
4) Write the relation in function notation. 5) What is the input? 6) What is the output? 7) Select ordered pairs: determine the average rate of change 8) What are the units of the average rate of change? 9) What is the practical meaning of the sign (+/-) of average rate of change? 0) Select other ordered pairs: determine the average rate of change ) Select other ordered pairs not used in #7 and #0: determine the average rate of change
) Using our results from questions #7, #0, and #, what can ou infer about the average rate of change for an interval? Average Rate of change: a comparison between the change in output values to the change in the input values using a ratio. change in output m change in input Linear Function: an function where the average rate of change between an pair of points is constant. 3) Is the value V of the car a linear function of the number of ears of ownership n? Explain using the definition of a linear function.
Treadmill times to walk, jog, or run 3 miles has been graphed as a function of weeks on an exercise program. 3- mile Time (minutes) Time in Program (weeks) 4) Describe in words what the graph is saing. 5) Calculate the Average Rate of change:
Slope: the average rate of change of a linear function. m m change in output change in input x x x 5) What is the significance of the numbers and in the formula above?
3- mile Time (minutes) Time in Program (weeks) Vertical intercept (-intercept), the point where the graph crosses the vertical axis. It alwas has (what value?) as the x-value of the point. Numericall it is alwas: (0, b) or (0, ) In function notation it is alwas: f ( 0) intercept 6) What is the practical meaning of the -intercept for the graph above?
V is the value of the car after n ears of ownership n (ears) 0 3 5 8 V, ($) 0,905 9,55 7405 5,655,55 6905 7) Once the initial value of the car was known, how did ou calculate the value for the other ears? 8) Write an equation for the function. V ( n) 0,905 750n The slope of the graph was the average rate of change (the earl depreciation rate): m = -750 V ( n) 0,905 750n The -intercept (the value of the car at n = 0) was: (0, 0,905) V ( n) 0,905 750n