Linear Relationships

Size: px

Start display at page:

Download "Linear Relationships"

Hester Evans
5 years ago
Views:

1 Linear Relationships Curriculum Read

3 Basics Page questions. Draw the following lines on the provided aes: a Line with -intercept and -intercept -. The -intercept is ( 0and, ) the -intercept is ( 0, -) b Line with -intercept and -intercept -. The -intercept is ^0, h and the -intercept ^-, 0h b b intercept (-,0) b intercept (0,) a intercept (,0) a intercept (0,-) - a - -. Write the following in gradient -intercept form: Gradient -intercept form is m+ c form. Rearrange each equation into this form. a + b % Linear Relationships Mathletics 00% P Learning SERIES TOPIC

4 Basics Page questions c d or or - Page questions. Write in standard form the equation for a line with gradient m and -intercept b Standard form is m+ c, where m is the gradient and c the -intercept. From the question m and c so the standard form of the equation is +.. Write the following in general form: General form is a + b + c 0. Rearrange each equation into this form: a b c d Find the gradient of the line given b + (Hint: Write in standard form first) Once the line is in standard form, the gradient (m) can be found b inspection and is in this case.. Write the equation for a line with -intercept b - and gradient m in general form. The gradient and -intecept are given, so first write the equation in standard form ( m+ c). Then rearrange into general form ( a + b + c 0) % Linear Relationships SERIES TOPIC Mathletics 00% P Learning

5 Knowing More Page 7 questions. Which of the points (-, ) or (-,) lies on the line - +? Both points have the same -value of -. Substitute the -value into the equation of the line and see which point 'works' for the line ( - ) + + The point (-, ) works for the equation, as the -value produced b the equation matches the -value of the point. It is then clear that the other point will not work, as we have just shown that - produces not. The point (-, ) is on the line Find an possible values for and if the point (, ) lies on the line + 7. Choose an -value ou like. I am choosing 00. so the point ( 00, 07 ) works for the line ( ) We can check b substituting the values back into the equation ( 00) The point ( 00, 07 ) works for the line + 7, which shows that the arithmetic is correct. 00% Linear Relationships Mathletics 00% P Learning SERIES TOPIC

6 Knowing More Page questions. Find an possible values for and if the point (, ) lies on the line Choose an -value ou like. I am choosing. First we need to rearrange to make the subject. So (, ) is a point on the line ( )- - We can check this b substituting the values back into the original equation: () - () Where we see that the equations 'work' for the point (,, ) so the arithmetic is correct.. Solve for if the point (,9) lies on the line Rearrange the equation to make the subject: and then substitute 9: ( ) So the point ( 9, ) is on the line % Linear Relationships SERIES TOPIC Mathletics 00% P Learning

7 Knowing More Page 9 questions. Are these lines parallel? Parallel lines have the same gradient. Rewrite in standard form ( m+ c) and compare gradient ( m ) terms. a + and These lines are not parallel as the first gradient (- ) is different from the second ( ). b + and These lines are not parallel as the first gradient ( ) is different from the second (- ). c - and These lines are not parallel as the first gradient ( ) is different from the second (- ). d and These lines are parallel as both gradients are the same (). 00% Linear Relationships Mathletics 00% P Learning SERIES TOPIC

8 Knowing More Page 9 questions. Find the value of if the line passing through (,0) and (,) is parallel to + 7. Parallel lines have the same gradient. The gradient of the + 7 line is. For the second line through ( 0, ) and (, ), use the formula for the gradient of a line between two points and set this equal to. - m ( - ) ( 0 - ) If a line has -intercept and is parallel to --, then what is the equation of the line? Let the new line be m+ c where m is the gradient and c in the -intercept. m must be - to be parallel with --. c must be as the -intercept is. The equation of the line is then % Linear Relationships SERIES TOPIC Mathletics 00% P Learning

9 Using Our Knowledge Page questions. Draw the following graphs using the table method: a (,) ^00, h ^-, -h b (, ) ^0, -h ^-0.,-. h % Linear Relationships Mathletics 00% P Learning SERIES TOPIC 7

10 Using Our Knowledge Page questions c ^-, h 7 (, 7) ^0, h d (,-) ^-, -h - - ^0, -h % Linear Relationships SERIES TOPIC Mathletics 00% P Learning

11 Using Our Knowledge Page questions. Draw the following lines on a number plane: a b - c d Page questions. Write the equations of the following lines: a c b (-,) (,) d - (-,-) - (, -) a - b c d -. Write down the coordinates where lines a and d - from the above question - intersect each other. a and d intersect at (-,-) 00% Linear Relationships Mathletics 00% P Learning SERIES TOPIC 9

12 Using Our Knowledge Page questions. Find the equations of the following lines: A vertical line passing through points (-, ) and (-,- ) : - A horizontal line passing through ( 0:, ) A line parallel to the -ais and passing through ( :, ) A line parallel to the -ais and passing through (-, ): - (-,) ^0, h ^, h ^-, -h - ^-, -h % Linear Relationships SERIES TOPIC Mathletics 00% P Learning

13 Using Our Knowledge Page questions. Draw the following graphs on the same set of aes: a - b c + d (,) (-,) (0,) (,) (-,) (0,0) (,-) (-,-) - (,-) What do ou notice about the lines as the value of m increases in their equations? If m is positive, the line moves from bottom left to top right If m is negative, the line moves from bottom right to top left 00% Linear Relationships Mathletics 00% P Learning SERIES TOPIC

14 Using Our Knowledge Page 7 questions. Draw these lines on the same set of aes below: a b 7- c (,7) (,) (,) (0,) (0,0) (-,-) - - ^0, -h (-,-) (-,-9) 00% Linear Relationships SERIES TOPIC Mathletics 00% P Learning

15 Thinking More Page questions. Find the intercepts of these lines: a - This equation is in standard form ( m+ c) so the -intercept is found b reading off the c value, -. Set 0 to find -intercept: 0 The -intercept is ( 0, -) and -intercept is (,0). - - b Set 0 to find -intercept: Set 0 to find -intercept: The -intercept is (- 7, 0) and -intercept is ( 0, - 7) c Set 0 to find -intercept: Set 0 to find -intercept: (0) (0) The -intercept is (- 9,0) and -intercept is (0, ). 00% Linear Relationships Mathletics 00% P Learning SERIES TOPIC

16 Thinking More Page questions d -- 0 Set 0 to find -intercept: -0 ( )- 0 Set 0 to find -intercept: The -intercept is (, 0 ) and -intercept is ( 0, - 7).. Draw the graph of a line with intercepts: a -intercept - and -intercept (0,) (-, 0) b -intercept and -intercept - ( 0, ) (0,-) % Linear Relationships SERIES TOPIC Mathletics 00% P Learning

17 Thinking More Page questions. Use the intercept method to sketch the graphs of the following equations: a -9 The equation is in m+ c form so the -intercept, - 9 can be seen directl. The -intercept is found b setting 0 and rearranging to find : The -intercept is at ( 0., ) 0 ^0, h ^0, -9h 00% Linear Relationships Mathletics 00% P Learning SERIES TOPIC

18 Thinking More Page questions b Set 0 to find -intercept: The -intercept is at ( 0, - ). Set 0 to find -intercept: The -intercept is at (-., 0) (0) (0) (-.,0) (0,-) - 00% Linear Relationships SERIES TOPIC Mathletics 00% P Learning

19 Thinking More Page questions. Graph each pair of lines on the same ais to find the point of intersection: a and - As the second line is in standard form we can see that the -intercept is at ( 0, - ). Set 0 to find -intercept: The -intercept is at (,0). 0 - ^0, h ^0, - -h -0 From the graph, the point of intersection is ( 0., ) 00% Linear Relationships Mathletics 00% P Learning SERIES TOPIC 7

20 Thinking More Page questions b - and Inspecting - shows the -intercept is at ( 0, - ). Set 0 to find -intercept: The -intercept is at (,0) For set 0 to find the -intercept: The -intercept is at (-,0). Set 0 to find -intercept: The -intercept is at ( 0, - ) ( ) (-,0) (,0) (0,-) (0,-) -, ` - j From the graph, we can see the point of intersection is (,- ). 00% Linear Relationships SERIES TOPIC Mathletics 00% P Learning

21 Thinking More Page questions. Find the point of intersection without drawing an graphs: a - and -- 0 As is alread the subject of the first equation, substitute the first equation into the second and solve for (-) From the first equation we know that -, so the point of intersection is ( 7, - ). b - and As is alread the subject of the first equation, it is eas to substitute the first equation into the second, and solve for ( -) As is alread the subject of the first equation, substitute 9 into the first equation to find. - 9 ` j- - The point of intersection is 9, ` j. 00% Linear Relationships Mathletics 00% P Learning SERIES TOPIC 9

22 Thinking More Page questions. Draw the following lines on the same set of aes: Line : + Line : - Line : - + Find intercepts for each line. All three lines are in standard from, so the -intercepts can be found b inspection. Line : + The line -intercept is at ( 0., ) Set 0 to find -intercept: Line -intercept is at (- 0, ). Line : - The line -intercept is at (0,- ). Set 0 to find -intercept: Line -intercept is at (,0) Line : - + The line -intercept is at (0, ). Set 0 to find -intercept: Line -intercept is at (,0). 0 00% Linear Relationships SERIES TOPIC Mathletics 00% P Learning

23 Thinking More Page questions (0,) (0,) (0.,7.) (-,0) (,0) (,0) (0,-) - + a What is the point of interception of Line and Line? Substitute Line into Line and solve for : Substitute this -value into Line to find : 0. The point of intersection of Line and Line is ( 0., 7. ) (. ) b Will Line and Line intersect at an point? No. c Wh do ou think this is so? Line and Line have the same gradient. This means the are parallel and will not intersect at an point. 00% Linear Relationships Mathletics 00% P Learning SERIES TOPIC

24 Thinking More Page questions 7. Identif whether the following pairs of lines will intersect or not. a + and -7 These have the same gradient ( ) so the will not intersect. b + and --7 These have different gradients ( and- ) so the will intersect. c + 7 and + Change the first line to standard form so that the gradient can be seen: The gradients are different (- and) so the lines will intersect. d and Rearrange both lines to standard form to see the gradients: These lines have the same gradient - so the will not intersect. 00% Linear Relationships SERIES TOPIC Mathletics 00% P Learning

25 Thinking More Page questions. Use substitution to prove that + and - intersect at the point (,). (Hint: Show the point of intersection is on both lines) Substitute into first line: This shows the point (, ) is on the first line. + ( ) + + Substitute into the second line: - () - - This shows the point (, ) is on the second line. If the same point is on different lines, then it must be the intersection point of the lines. 9. What is the point of intersection of the lines - and 7? The first line is horizontal with all -value being -. The second line is vertical with all -values being 7, so the point of interaction is (7, - ). 0. Find the equation of the horizontal line, eactl in the midle of - and. The new line will also be horizontal, so it will be a something line. Use an average to find a number that is half wa between - and : average - + The line eactl in the middle of - and is. You can check this b noting that is below, and above -. 00% Linear Relationships Mathletics 00% P Learning SERIES TOPIC

26 Notes 00% Linear Relationships SERIES TOPIC Mathletics 00% P Learning

27 Notes 00% Linear Relationships Mathletics 00% P Learning SERIES TOPIC

28 Notes 00% Linear Relationships SERIES TOPIC Mathletics 00% P Learning

30 Linear Relationships

Using Intercept Form

8.5 Using Intercept Form Essential Question What are some of the characteristics of the graph of f () = a( p)( q)? Using Zeros to Write Functions Work with a partner. Each graph represents a function of