KS3: Straight Lines

KS3: Straight Lines Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 27th August 2013

y What is the equation of this line? And more importantly, why is it that? 4 3 2 1 -1 -2 -3 -4 x-5-4 -3 -2 -1 0 1 2 3 4 5 6 ?

y What and why? 4 3 2 1 -1 -2 -3 -4 x-5-4 -3 -2 -1 0 1 2 3 4 5 6 ?

y Sketch 4 3 2 1 -1 -2 -3 -4 x-5-4 -3 -2 -1 0 1 2 3 4 5 6 Do you notice any connection between how increases each time and the equation? ? ? ? ?

y Sketch 4 3 2 1 -1 -2 -3 -4 x-5-4 -3 -2 -1 0 1 2 3 4 5 6 ? ? ? ?

! ? The steepness of a line is known as the gradient. It tells us what changes by as increases by 1. The equation of a straight line is of the form: The gradient is . Gradient

y 4 D A 3 F C 2 B 1 E G x-5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 H -2 -3 On your printed sheet, identify the gradient of each line. -4

y 4 D A 3 F C 2 B 1 E G x-5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 H -2 -3 -4

y Suppose we just had two points on the line and wanted to determine the gradient, but didn’t want to draw a grid. 4 3 has increased by 6. 2 1 x-5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 -2 has increased by 4. -3 ? So what does change by for each unit increase in ? -4

Gradient using two points ! Given two points on a line, the gradient is: ? ? ?

Gradient using two points Work out the gradient given the points on the line. Point 1 Point 1 Gradient Q1 (0,0) (2,6) ? 3 Q2 (1,3) (3,7) 2 ? Q3 (0,5) (4,25) 5 ? Q4 (2,2) (-1,5) ? -1 Q5 (4,3) (10,6) 0.5 ? Q6 (7,8) (-4,-3) 1 ? Q7 (7,1) (-1,5) ? -0.5 Q8 (6,5) (8,1) -2 ? Q9 (1,3) (5,10) 1.75 or 7/4 ? Q10 (-1,4) (9,-5) ? -0.9 or -9/10 Q11 (1,0) (-2,-4) ? 4/3 Q12 (1.5, -6.5) (-1.75 , 4.3) -3.323 to 3dp ?

Summary The gradient of a line is the steepness: how much changes as increases by 1. We’ve seen 3 ways in which we can calculate the gradient: a. Counting Squares b. Using the equation c. Using two points ? ? ?

y Recap 4 D A 3 F C 2 B 1 E G x-5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 H -2 -3 -4

y y-intercept 4 D A 3 F C 2 B 1 E G x-5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 H • The y-intercept is the point at which the line crosses the -axis. • It is the in (why?) -2 -3 -4

y 4 D A 3 F C 2 B 1 E G x-5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 H -2 -3 Now determine the full equation of each line. -4

Equations of lines Copy and complete the following tables. Easier: Harder: ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? This is ‘Part 2’ on your worksheet.

Card Sort!

Puzzle (This was in the Year 8 End of Year exam) The diagram shows three points and . A line is parallel to and passes through . Find the equation of the line . ?

Now you try on your own… The diagram shows three points and . A line is parallel to and passes through . Find the equation of the line . ?

Puzzle (This was in the Year 8 End of Year exam) Find the coordinates of the points where the line cuts: The -axis: The -axis: ? ?

Finding axis intersection Copy and complete this table. The point where the line crosses the: ? ? ? ? ? ?

Exercises (Part 6) Give the equation of a line which is parallel to y = 3x + 2. y = 3x + c (where c is any positive or negative number!) Give the equation of a line which passes through (0, -2) and is parallel to another line which passes through the points (1, 4) and (5, 24) y = 5x – 2 Give the equation of a line which passes through the point (0, 6) and has the gradient -2. y = -2x + 6 Which line has the greater gradient, 4x – 5y = 1, or 5x – 4y = 1 First line rearranges to y = 0.8x – 0.2, second to y = 1.25x – 0.25 So second line has the greater gradient. 1 ? 2 ? 3 ? 4 ? Give the coordinate of the point where the following lines crosses the x-axis: y = -2x + 4 (2, 0) y = 0.5x – 4 (8,0) x + y = 5 (5,0) 2x + 3y = 4 (2,0) Give the coordinate of the point where the following lines crosses the y-axis: y = 2x – 3 (0, -3) y = 1 – 3x (0,1) x + y = 5 (0,5) 2x + 3y = 4 (0, 4/3) 5 6 ? ? ? ? ? ? ? ?

Equation given a gradient and point The gradient of a line is 3. It goes through the point (4, 10). What is the equation of the line? • Start with (where is to be determined) • Substituting: • Therefore ? The gradient of a line is -2. It goes through the point (5, 10). What is the equation of the line? ?

Exercises Gradient Goes through Full equation Q1 3 (4,5) ? Q2 5 (2,3) ? Q3 -1 (2,5) ? Q4 2 (4,-2) ? Q5 (10,11) ? Q6 (-9,4) ? ? Q7 (6, 6)

Equation given two points A straight line goes through the points (3, 6) and (5, 12). Determine the full equation of the line. Gradient: 3 Equation: ? (5,12) ? (3,6) A straight line goes through the points (5, -2) and (1, 0). Determine the full equation of the line. (5, -2) Gradient: -0.5 Equation: ? (1,0) ?

Exercises Work out the gradient given the points on the line. Point 1 Point 2 Full Equation Q1 (0,0) (2,2) ? Q2 (-5,0) (0,-5) ? Q3 (1,-3) (3,1) ? Q4 (-4,1) (4, 5) ? Q5 (-3,7) (2,2) ? Q6 (1,6) (3,-2) ? Q7 (-7,3) (5,-1) ? ? Q8 (4,9) (-3,10)

REVISION Vote with your diaries!

The equation of a line is . What is the missing value of this point on the line?

y 4 3 2 1 x-5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 -2 -3 -4

What is the equation of a line parallel to and goes through the point ?

What is the gradient of the line which goes through the points and ?

What is the full equation of a line which has gradient 3 and passes through the point (2,5)?

What is the full equation of the line which goes through the point , ?

What is the y-intercept of the line ?

What is the gradient of the line ?

Give the coordinate of the point where the line crosses the axis.

KS3: Straight Lines