Straight Lines. Straight Lines. Curriculum Ready.

Transcription

1 Curriculum Read

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3 Straight lines all follow a particular pattern or rule and appear in all facets of life. This curv optical illusion is made using lots of lines that have different slopes. Give this a go! Q The robot standing on the at point A needs to get to point B on the. The solar panels onl have enough stored energ to travel the shortest straight line path. Write down the rule of the line the robot needs to follow to get from A to B. B A Work through the book for a great wa to solve this P Learning I SERIES TOPIC

4 How does it work? Graphs using tables of values Let s review graphing lines from a completed table of values. Values in the table are paired together to find the coordinates. The table below was completed using the rule: = x- x (-, -) (-, -) (-, -) (, -) (, -) (, ) Write the rule along the line Remember: The x-value is alwas written first = x Join plots with a straight, double arrowed line Plot tabled values Summar for graphing from a table of values: Plot the coordinates read from the table as small dots Use a ruler to join the points and put neat arrows on either end of the line Write the rule used along the line I SERIES TOPIC P Learning

5 How does it work? Your Turn Graphs using tables of values Plot each of the completed table of values below: = x = - x x x GRAPHS USING TABLES OF VALUES.../.../... TABLES OF VALUES GRAPHS USING = x+ = -x x x P Learning I SERIES TOPIC

6 How does it work? Completing and graphing tables of values Sometimes ou are onl given a rule and need to complete our own table of values. The table of values often needs to be completed using the given rule before plotting the graph. Complete and plot the graph for the table of values using the rule = x+ = x+ x = ( - ) + = - = ^- h+ = ^ h+ = ^ h+ = ^h+ = - = = = Alwas include zero, positive and negative values in our table of values Write the equation along the line = x + Plot tabled values Join plots with a straight, double arrowed line T-Charts Tables of values can also be written verticall. When drawn this wa the are often called T-charts. x - Coordinates are paired horizontall I SERIES TOPIC P Learning

7 GRAPHING AND COMPLETING TABLES OF VALUES * How does it work? Your Turn Completing and graphing tables of values Complete these tables of values for each given rule and then plot the graph. a = x- b =-x = x+.../.../... x - - x c x =- + d x = + x - - x P Learning I SERIES TOPIC

8 How does it work? Your Turn Completing and graphing tables of values Complete these tables of values for each given rule and then plot the graphs. = x- =- x+ a b x - - x c = - x d x - - = x x I SERIES TOPIC P Learning

9 How does it work? Your Turn Completing and graphing tables of values Complete the T-charts below using the given rule and then plot the graphs. a = x- x b =-x- x c = x + x P Learning I SERIES TOPIC

10 How does it work? Pattern of movement The vertical (up & down) and horizontal (left & right) movement from one point to the next on a straight line follows a pattern. This remains the same (is constant) all along the line. Describe the constant pattern of movement for this linear graph: Remember: Linear means straight line Up unit Across unit Vertical and horizontal construction lines ` Constant pattern of movement is: Up one unit (+) verticall for ever one unit (+) to the right horizontall + or -? For vertical movement, up is positive (+ ) and down is negative (- ) Because we alwas read left to right, the horizontal movement is alwas positive Describe the constant pattern of movement for the linear graph below: Down unit Across units ` Constant pattern of movement is: Down one unit (-) verticall for ever three units (+) to the right horizontall 8 I SERIES TOPIC P Learning

11 How does it work? Your Turn Pattern of movement Describe the constant pattern of movement for each of the following linear graphs. a b /.../ * PATTERN OF MOVEMENT + PATTERN OF MOVEMENT * Pattern of movement: Pattern of movement: verticall for ever across verticall for ever across c d Pattern of movement: Pattern of movement: verticall for ever across verticall for ever across e f Pattern of movement: Pattern of movement: verticall for ever across verticall for ever across P Learning I SERIES TOPIC 9

12 How does it work? Your Turn Pattern of movement Draw a linear graph that matches these constant patterns of movement. a + verticall for ever + horizontall b + verticall for ever + horizontall c - verticall for ever + horizontall d - verticall for ever + horizontall e - verticall for ever + horizontall f verticall for ever value horizontall I SERIES TOPIC P Learning

13 How does it work? Horizontal and vertical lines These lines are special because the pattern of movement changes in one direction onl. Horizontal Lines The rule is: = a constant number Graph is parallel to the Parallel lines never cross Write the rule of the graph below: The constant pattern of movement is: verticall for ever value horizontall ` The -value remains unchanged with a value of, regardless of the x-value ` equation is: = Vertical Lines The rule is: x = a constant number Graph is parallel to the Write the equation of the graph below: The constant pattern of movement is: horizontall for ever value verticall ` The x-value remains unchanged with a value of, regardless of the -value ` equation is: x = P Learning I SERIES TOPIC

14 + How does it work? Your Turn Horizontal and vertical lines Write the rule for these graphs. a b HORIZONTAL AND VERTICAL LINES.../.../... HORIZONTAL AND VERTICAL LINES + Rule: Rule: c d Rule: Rule: Sketch the following graphs: a =-. b x = I SERIES TOPIC P Learning

15 How does it work? Intercepts The exact point where a graph crosses an axis is called an intercept. Intercepts are named using the axis the cross. For the equation = x+ : (i) Write down the intercepts of the graph for = x+ x-intercept: The intercept on the horizontal axis = x+ -intercept: The intercept on the vertical axis ` x-intercept = - and Intercept points with axes will have at least one zero coordinate value. (ii) Write down the coordinates for these intercepts x-intercept = (-, ) (, ) Intercept points can be found b substituting zero into the rule for each variable. (iii) Use the rule = x+ to find the intercepts another wa -intercept is where x = = x+ = # + = ` -intercept is (, ) x-intercept is where = = x+ = x + - = x + - x =- ` x-intercept is (-, ) substitute x = into the rule substitute = into the rule and solve for x P Learning I SERIES TOPIC

16 How does it work? Let s tr another one! For the equation x+ - = : (i) Write down the intercepts of the graph for x+ - = This wa is called -intercept general form x+ - = ` x-intercept = and x-intercept -intercept = Intercept points with axes will have at least one zero coordinate value. (ii) Write down the coordinates for these intercepts x-intercept = (,) (, ) Intercept points can be found b substituting zero into the rule for each variable. (iii) Use the rule x+ - = to find the intercepts another wa -intercept is where x = + - = substitute x = into the rule = ' = ' = ` -intercept is (, ) x-intercept is where = # + x - = x - = x - + = + x = ` x-intercept is (,) substitute = into the rule Remember: For an point on the, x = ` -intercept is found b substituting x = into the rule For an point on the, = ` x-intercept is found b substituting = into the rule I SERIES TOPIC P Learning

17 How does it work? Your Turn Intercepts Write down the coordinates of the x and intercepts for the following graphs. a b c x-intercept = x-intercept = x-intercept = d e f x-intercept = x-intercept = x-intercept = Horizontal & vertical lines l Horizontal lines onl intercept the -intercept x l Vertical lines onl intercept the x x-intercept P Learning I SERIES TOPIC

18 How does it work? Your Turn Intercepts Write down the coordinates of the x and intercepts for the graphs of the following straight line rules. hint: look at part (iii) of the examples for the method a = x+ b x x-intercept = = + x-intercept = c = x+ d x x-intercept = = - x-intercept = e + x = f x x-intercept = - = x-intercept = I SERIES TOPIC P Learning

19 COMBO TIME * COMBO TIME * COMBO TIME * How does it work? Your Turn Combining intercepts with the pattern of movement Write down the coordinates of the other intercept point using the given information. hint: On the number plane, start from the given point and use the pattern of movement to help draw the line first x-intercept is (, ), -intercept is? Constant pattern of movement is + verticall for ever + horizontall intercept is (, ), x-intercept is? Constant pattern of movement is + verticall for ever + horizontall intercept is (-, ), x-intercept is? Constant pattern of movement is - verticall for ever + horizontall..../.../ P Learning I SERIES TOPIC

20 Where does it work? Graphing using the intercepts A linear graph can be drawn from a rule b plotting and joining the intercepts. Draw the graph of - x = b finding and plotting the intercepts first For - x = : -intercept is where x = - x = - # = = ` -intercept is (, ) x-intercept is where = - x = - x = x = ' - x =- ` x-intercept is (-, ) substitute x = into the rule substitute = into the rule Step : Plot the intercepts Step : Write the rule along the line (-, ) x = (, ) Step : Draw a double-arrowed line through both points 8 I SERIES TOPIC P Learning

21 Where does it work? Draw the graph of =- x+ b finding and plotting the intercepts first For =- x+ : -intercept is where x = =- x+ =- # + = ` -intercept is (, ) x-intercept is where = =- x+ =- x + x = x = ` x-intercept is (,) substitute x = into the rule substitute = into the rule (, ) Step : Plot the intercepts (,) Step : Draw a double-arrowed line through both points Step : Write the rule along the line =- x+ P Learning I SERIES TOPIC 9

22 * GRAPHING USING THE INTERCEPTS * GRAPHING USING THE INTERCEPTS Where does it work? Your Turn Graphing using the intercepts Graph each of the following rules using the intercepts method. x + = x-intercept = /.../... + x = x-intercept = x = x-intercept = I SERIES TOPIC P Learning

23 Where does it work? Your Turn Graphing using the intercepts Graph each of the following rules using the intercepts method. - x = 8 x-intercept = = x- x-intercept = x = x-intercept = P Learning I SERIES TOPIC

24 Where does it work? Slope The slope of a line is the special name given to the constant pattern of movement written as a fraction. Positive Vertical movement Horizontal movement Also called slope or Rise Run If the line slopes up the slope is positive. What is the slope of the line graphed below? Up unit sloping up = positive slope Across unit Pattern of movement is: + (up) verticall for ever + across ` = Negative If the the line slopes down the slope is negative. What is the slope of the line graphed below? Down unit Across units sloping down = negative slope Pattern of movement is: - (down) verticall for ever + across ` - =- I SERIES TOPIC P Learning

25 Where does it work? Using the slope formula on horizontal and vertical lines gives special results. Zero Horizontal lines have a slope of. What is the slope of the horizontal line graphed below? No vertical change Pattern of movement is: verticall for ever horizontal value ` = an x-value an values across Undefined The slope for vertical lines cannot be defined. Let s see wh with this example. What is the slope of the vertical line graphed below? An values verticall No horizontal change Pattern of movement is: ever vertical value for horizontall ` an -value = undefined because ou cannot divide b As the value of the slope gets bigger, the graph gets steeper. P Learning I SERIES TOPIC

26 * GRADIENT * GRADIENT * GRADIENT * GRADIENT * GRADIENT Where does it work? Your Turn Slope RA SLOPE * SLOPE * SLOPE * SLOPE * SLOPE * SLOPE * Rise (+) (i) What is the slope of the lines graphed below? (ii) Circle the graph for each pair that has the steeper slope. Run.../.../... a b c d I SERIES TOPIC P Learning

27 Where does it work? Your Turn Slope What is the slope of these lines? a b c d e f P Learning I SERIES TOPIC

28 Where does it work? Your Turn Slope Sketch a line that has the given slope on the number planes below. a Slope = b Slope = c Slope =- d Slope = e Slope = f undefined g Slope =- h Slope = I SERIES TOPIC P Learning

29 What else can ou do? Graphing straight lines using the slope-intercept rule The slope and -intercept can be found easil from a straight line rule when written a special wa. mx b = + the number in front of x (m) is the slope the constant term (b) is the -intercept For the rule = x+ : (i) Write the slope and -intercept for the graph of = x+ = x+ All whole numbers can be written as fractions (ii) Graph = x+ Slope ^mh = = + -intercept ^b h =+ + ` Graph of the rule moves + verticall for ever + horizontall and passes through the at (, ) Step : Plot the -intercept Step : Use the slope to plot a second point The number in front of the x in the rule is called the coefficient of x = x+ Step : Draw a double arrowed line through the points Step : Write the rule along the line P Learning I SERIES TOPIC

30 What else can ou do? Here is another example with a fraction written in the rule: For the rule =- x+ : (i) Write the slope and -intercept for the graph of =- x+ =- x+ Slope ^mh =- = - -intercept ^bh =+ + ` Graph of the rule moves - verticall for ever + horizontall and passes through the at (, ) Rules like = - x+, can also be written as - x = - x = - x + (ii) Graph =- x+ Step : Plot the -intercept (-, ) Step : Use the slope to plot a second point Step : Write the rule along the line = - x Step : Draw a double arrowed line through the points Sometimes we need to extend an axis to find an intercept 8 I SERIES TOPIC P Learning

31 * GRAPHING STRAIGHT LINES USING THE GRADIENT-INTERCEPT RULES What else can ou do? Your Turn Graphing straight lines using the slope-intercept rule (i) Write the slope and -intercept for each straight line rule. (ii) Sketch each graph. = m x + b GRAPHING STRAIGHT LINES USING THE SLOPE-INTERCEPT RULE *.../.../... a = x- b x = c = x- d x = e =- x+ or = - x f = x P Learning I SERIES TOPIC 9

32 What else can ou do? Your Turn Graphing straight lines using the slope-intercept rule (i) Write the slope and -intercept for the graph of each straight line rule. (ii) Sketch each graph. a = x+ b =- x c = x- or = x d =- x e = x + f = x Be careful finding the second point I SERIES TOPIC P Learning

33 What else can ou do? Finding the slope-intercept rule from the graph We reverse the method of graphing to find the rule of a linear graph in slope-intercept form. Find the rule of these graphed lines ` = x+ -intercept ^bh =+ Slope ^ m h = mx+ b Step : Read the -intercept + Step : Determine the slope in simplest form = + = + = + + (-, ) is a point with clear coordinates You might need to look a little more closel to find another point with clear coordinates to use for slope. -intercept = Step : Read the -intercept Step : Determine the slope in simplest form -intercept ^bh =- Slope = mx+ b ` = - x - ^ m h = - =- + P Learning I SERIES TOPIC

34 * FINDING THE GRADIENT-INTERCEPT RULE FROM THE GRAPH FINDING THE SLOPE-INTERCEPT RULE FROM THE GRAPH * What else can ou do? Your Turn Finding the slope-intercept rule from the graph Find the rule of the line for each graph below: = x - l.../.../... a b ` Rule: ` Rule: c d ` Rule: ` Rule: e f ` Rule: ` Rule: I SERIES TOPIC P Learning

35 What else can ou do? Your Turn Finding the slope-intercept rule from the graph The -intercept, slope or both for each of these trickier questions are not whole numbers. a b ` Rule: ` Rule: c d ` Rule: ` Rule: e f ` Rule: ` Rule: P Learning I SERIES TOPIC

36 What else can ou do? Your Turn Graphing straight lines using the slope-intercept rule The robot standing on the at point A needs to get to point B on the. The solar panels onl have enough stored energ to travel the shortest straight line path. Write down the rule of the line the robot needs to follow to get from A to B. hint: determine the -intercept and slope and go from there. B A Remember me? Rule of the line for the robot to follow: Two kes are needed to unlock a treasure chest at Canar Cove. Bird Town and Flutterton have one ke each. Two homing pigeons, one from each town, are sent to Canar Cove. Use the map below to find the straight line rule each pigeon needs to fl along to get to Canar Cove. hint: draw the flight paths of each pigeon first Canar Cove Bird Town Flutterton Rule of the line for the pigeon from Bird Town: Rule of the line for the pigeon from Flutterton: I SERIES TOPIC P Learning

37 What else can ou do? Your Turn Reflection Time Reflecting on the work covered within this booklet: What useful skills have ou gained b learning about straight lines? Write about one or two was ou think ou could appl straight lines to a real life situation. If ou discovered or learnt about an shortcuts to help with straight lines or some other cool facts, jot them down here: P Learning I SERIES TOPIC

38 Cheat Sheet Here is a summar of the important things to remember for straight lines Graphing from a table of values The x-value is alwas written first in coordinates: (x-value, -value) = (horizontal value, vertical value) Pattern of movement The pattern of vertical and horizontal movement for a straight line does not change (is constant). Horizontal and vertical lines Horizontal: rule: = a constant value Vertical: rule: x = a constant value Intercepts x-intercept (where = ) -intercept (where x = ) To graph a rule using the intercepts, plot the intercept points first and then join with a straight line. Slope Vertical movement Horizontal movement = rise run = positive slope, = negative slope, = zero slope, = slope undefined The larger the value of the slope, the steeper the line: = large slope value = small slope value Slope-intercept rule mx b = + coefficient of x (m) is the slope the constant term (b) is the -intercept To draw a graph using this rule: plot the -intercept use the slope to plot a second point join the two points with a straight line (putting arrows on each end). To find the rule from a graph: write down the -intercept find another point on the line with clear coordinates find the slope between this point and the -intercept put the -intercept and slope values into the slope-intercept rule. I SERIES TOPIC P Learning

39 Notes P Learning I SERIES TOPIC

40 Notes 8 I SERIES TOPIC P Learning

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