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GCSE: Straight Line Equations

GCSE: Straight Line Equations. Dr J Frost (jfrost@tiffin.kingston.sch.uk ) www.drfrostmaths.com. Last modified: 30 th August 2015. GCSE specification:. Understand that an equation of the form y = mx + c corresponds to a straight line graph Plot straight line graphs from their equations

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GCSE: Straight Line Equations

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  1. GCSE: Straight Line Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com Last modified: 30th August 2015

  2. GCSE specification: • Understand that an equation of the form y = mx + c corresponds to a straight line graph • Plot straight line graphs from their equations • Plot and draw a graph of an equation in the form y = mx + c • Find the gradient of a straight line graph • Find the gradient of a straight line graph from its equation • Understand that a graph of an equation in the form y = mx + c has gradient of m and a y intercept of c (ie. crosses the y axis at c) • Understand how the gradient of a real life graph relates to the relationship between the two variables • Understand how the gradients of parallel lines are related • Understand how the gradients of perpendicular lines are related • Understand that if the gradient of a graph in the form y = mx + c is m, then the gradient of a line perpendicular to it will be -1/m • Generate equations of a line parallel or perpendicular to a straight line graph

  3. 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 The line represents all points which satisfies the equation. • □ “Understand that an equation corresponds to a line graph.” ?

  4. y Starter 4 A D 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 What is the equation of each line?

  5. Equation of a line • Understand that an equation of the form corresponds to a straight line graph • The equation of a straight line is gradient y-intercept

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

  7. Gradient from an Equation • Find the gradient of a straight line graph from its equation. Putting in form : Gradient is Putting in form : Gradient is -2 ? ?

  8. Test Your Understanding Find the gradient of the line with equation . ?

  9. Exercise 1 Determine the gradient of the lines which go through the following points. Determine the gradient of the lines with the following equations: A line goes through the points and . Line has the equation . Which has the greater gradient: So has greater gradient. ? 1 d ? e ? ? a f ? b ? g c ? ? d 3 e ? f ? g ? ? ? h 2 a ? b ? c ?

  10. y Drawing Straight Lines 4 3 2 1 -1 -2 -3 -4 Sketch the line with equation: x-5-4 -3 -2 -1 0 1 2 3 4 5 6 • □ “Plot and draw a graph of an equation in the form y = mx + c” Bro Tip: To sketch a line, just work out any two points on the line. Then join up. Using for one point and for the other makes things easy.

  11. y Test Your Understanding 4 3 2 1 -1 -2 -3 -4 Sketch the line with equation: x-5-4 -3 -2 -1 0 1 2 3 4 5 6 • □ “Plot and draw a graph of an equation in the form y = mx + c”

  12. Finding intersection with the axis When a line crosses the -axis: When a line crosses the -axis: ? ? The point where the line crosses the: ? ? ? ? ? ?

  13. 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? ?

  14. Test Your Understanding Determine the equation of the line which has gradient 5 and goes through the point . Determine the equation of the line which has gradient and goes through the point . Find the equation of the line which is parallel to and goes through the point 1 ? 2 ? 3 ?

  15. 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) ?

  16. Exercise 2 Determine the points where the following lines cross the and axis. Using suitable axis, draw the line with equation . A line has gradient 8 and goes through the point . Determine its equation. A line has gradient and goes through the point . Determine its equation. Determine the equation of the line parallel to and goes through the point . Determine the equation of the line parallel to and goes through the point . Determine the equation of the lines which go through the following pairs of points: 5 1 ? ? ? 6 ? ? 2 7 ? ? ? ? ? 3 ? ? 4 ?

  17. y 4 3 2 1 -1 -2 -3 -4 m = -1/3 ? m = 1/2 ? m = 3 ? x-5-4 -3 -2 -1 0 1 2 3 4 5 6 m = -2 ? Find the gradients of each pair of perpendicular lines. What do you notice?

  18. Perpendicular Lines ! If two lines are perpendicular, then the gradient of one is the negative reciprocal of the other. Or: ? ? ? ? ? ?

  19. Example Problems Q1 A line is goes through the point (9,10) and is perpendicular to another line with equation . What is the equation of the line? A line goes through the points and . A second line is perpendicular to and passes through point B. Where does cross the x-axis? Are the following lines parallel, perpendicular, or neither? Neither. Gradients are and . But , not -1, so not perpendicular. ? Q2 ? Q3 ?

  20. Exercise 3 4 A line goes through the indicated point and is perpendicular to another line . Determine the equation of in each case. Find the equation of the line which passes through B, and is perpendicular to the line passing through both A and B. Line has the equation . Line goes through the points and . Are the lines parallel, perpendicular, or neither? so perpendicular. 1 ? ? Determine the equation of the line . ? ? ? 5 ? 2 ? Determine the equation of the line . Known point on : So equation of : 3 ? ?

  21. GCSE specification: • Understand that an equation of the form y = mx + c corresponds to a straight line graph • Plot straight line graphs from their equations • Plot and draw a graph of an equation in the form y = mx + c • Find the gradient of a straight line graph • Find the gradient of a straight line graph from its equation • Understand that a graph of an equation in the form y = mx + c has gradient of m and a y intercept of c (ie. crosses the y axis at c) • Understand how the gradient of a real life graph relates to the relationship between the two variables • Understand how the gradients of parallel lines are related • Understand how the gradients of perpendicular lines are related • Understand that if the gradient of a graph in the form y = mx + c is m, then the gradient of a line perpendicular to it will be -1/m • Generate equations of a line parallel or perpendicular to a straight line graph

  22. Two last things… Distance between two points Midpoint of two points ? 5 3 ? 4 Just find the average of and the average of . Find change and change to form right-angled triangle. Then use Pythagoras.

  23. Past Exam Questions See GCSEPastPaper_Solutions.pptx GCSERevision_StraightLineEquations.docx

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