LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0


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1 LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( ) = a + b. The graph of a linear equation is a graphical view of the set of all points that make the equation true. The graph of an linear function is a straight line. A linear function can be represented in two was, standard form and slopeintercept form. Standard form is a formal wa of writing a linear equation, while slopeintercept form makes the equation easier to graph. Form Equation Note Standard A + B = C A and B are not 0. A > 0 Slopeintercept = m + b m is the slope of the line and b is the intercept. To graph a linear function we must first identif the and intercepts. The intercept is the point where the graph crosses the ais and the intercept is the point where the graph crosses the ais. To find the intercept:. Set = 0 in the equation.. Solve for. The value obtained is the coordinate of the intercept. 3. The intercept is the point (, 0), with the value found in step. To find the intercept:. Set = 0 in the equation.. Solve for. The value obtained is the coordinate of the intercept. 3. The intercept is the point (0, ), with the value found in step.
2 Vertical Lines Equations of the form = a are vertical lines. The coordinate of ever point on the vertical line = a has the value "a," alwas, for an given value.. Horizontal Lines Equations of the form = a are horizontal lines. The coordinate of ever point on the horizontal line = b has the value "b," alwas, for an given value. Slope The slope of a line refers to the slant or inclination of the line. The slope is the ratio of the vertical change to the horizontal change between two points on the line. The slope can also be called the rise over run ratio because it tells ou how man spaces to move up or down and how man spaces to move to the right. A positive sign will move the line up and a negative sign will move the line down. One important thing to remember is that the run will alwas be to the right, regardless of the sign. m rise change in = = = run change in
3 The formula to find the slope of a line passing through the points ( ) and ( ),, is: m = Note: A horizontal line has slope of 0, while a vertical line has an undefined slope. Eample: Find the slope of the line: You can use an two points on a line to calculate its slope, our answer will be the same no matter which points ou choose. Choosing the points (, ) and (, 0): m = 0 = = = Parallel Lines In the intercept form equation, = m + b, m is the slope and b is the intercept. Two lines are parallel if their slopes are the same ( m = m ) and their intercepts are different ( b b ). Eample: ) = + 9 m = b = 9 ) = m = b = In the previous eample the slope of both equations is the same and their intercepts are different; therefore, the lines are parallel.
4 Perpendicular Lines Two lines are perpendicular if the slopes are the negative reciprocal of each other: m Eample: ) = + 7 m = b = 7 ) = + m = b = =. m The slope of equation is the negative reciprocal of the slope of equation ; therefore, the lines are perpendicular Finding the equation of a line To find the equation of a line when onl points or a slope is given, use the pointslope form of a linear equation formula: = m where m is the slope of the line and ( ) ( ), is a point on the line.
5 Eamples: Find the equation of the line which passes through the point (, ) and whose slope is 5. Using the pointslope form equation we obtain: m( ) ( ( ) ) = = 5 = = 5 + Resultant Equation Find the equation of the line through the points (, 5) and (, ). First find the slope between these two points using the slope equation: m = = 5 = = ( ) With the slope obtained and one of the two points given, use the pointslope form equation to find the equation of the line. = m ( ) 5 = ( ( ) ) 5 = + = + Resultant Equation Eventuall, ou ma be asked to find the equation of a line that is parallel or perpendicular to a given line. Just remember that the slopes of two parallel lines are eactl the same and that the slopes of two perpendicular lines are the negative reciprocals of each other. Eamples: Find the equation of a line that is parallel to = + and that passes through the point (, 7). = + Since the equation is in slopeintercept form, the coefficient of is m = the slope of the line. Remember =m+b (m is the slope!). Since the lines must be parallel, use the same slope and the given point to find the equation of the parallel line: m = m =
6 ( ) ( ) = m 7= 7= 8 = Parallel line to = +. Find the equation of a line perpendicular to + = 9and that passes through the point (, ). + = 9 This equation is in standard form, we need to convert it to slopeintercept form: =m+b. = = 9 = + The slope is m =. Since the second equation must be perpendicular to the first equation, find the negative reciprocal. m = m = = m Using the slope m = and the point (, ): = m ( ) ( ) = ( ( ) ) + = + = Perpendicular line to + = 9.
7 LINEAR FUNCTIONS EXERCISES. Identif the false statement below a) The standard form of a linear equation is = a + b b) The point (, 0) is on the graph of 3 = c) The graph of 3 = passes the vertical line test for functions d) The intercept of 3 = is (0, 3). Identif the false statement below a) To find the intercept of a graph, set equal to zero and solve the equation for b) The graph of = 3 is a horizontal line three units above the ais c) Generall speaking, it is a good idea to plot three points when constructing the graph of a linear function d) To find the intercept of a graph, set equal to zero and solve the equation for 3. The standard form of the equation 5 = 30 is a) + 5 = 30 b) = + 5 c) 5 = 30 d) 5 = 30. Identif the true statement below a) All linear graphs are functions b) The graph of = is a function c) To find the intercept of a function, let equal to zero and solve for d) The graph of = is a function 5. Determine the slope and intercept of each equation: a) + 7 = b) + 5= 0 c) 3 + 9= 0. Use the point slope form to find the equation of the line which passes through (, ) and whose slope is Write in standard form the equation of the line passing through the point ( 3, ) with slope equal to 3 m =. 8. Which of the following is the equation of a line in standard form passing through the point (3, 0) and perpendicular to the line =? a) + = b) + = 3 c) = d) = 3 9. Find the equation of a line in standard form that passes through the points (, ) and (3, ). 0. Determine whether the two lines are parallel, perpendicular or either:3 5 = and 0 = 7
8 LINEAR FUNCTIONS ANSWERS TO EXERCISES. A is the false statement. The standard form of a linear equation is A + B = C.. B is the false statement. The graph of = 3 is a vertical line 3 units to the right of the ais. 3. D: 5 = 30,A> 0. B: = is an eample of a linear function. 5. a) m= and b= 7 b) m = undefinded and b does not eist c) m= 0 and b= = = 8. B: + = 3 is a perpendicular line to = 9. + = 7 0. The lines are parallel.
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