GCSE: Straight Line Equations Dr

GCSE: Straight Line Equations Dr J Frost ([email protected]) www.drfrostmaths.com Last modified: 30th 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 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

y What is the equation of this line? And more importantly, why is it that? 4 3 2 1 x -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 -2 ? 𝑥 2 The line -3represents all points which satisfies -4the equation. “Understand that an equation corresponds to a line graph.” 6

y 4 Starter A D 3 F C 2 B 1 E G x -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 -2 H -3 -4 What is the equation of each line?

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

Gradient using two points ! Given two points on a line, the gradient is: ( 1 , 4 ) (3 , 10) ( 5 , 7) ( 8 ,1) ( 2 , 2 ) ( 1 , 10) 𝑚 ?3 ? 𝑚 2 8 𝑚 ? 3

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

Test Your Understanding Find the gradient of the line with equation . 𝟐𝒚 𝒙 𝟏 ?

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

Drawing Straight Lines y 4 Sketch the line with equation: 3 2 1 x -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 -2 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. -3 -4 “Plot and draw a graph of an equation in the form y mx c” 6

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

Finding intersection with the axis 𝑦 When a line crosses the -axis: ? When a line crosses the -axis: ? 𝑥 The point where the line crosses the: Equation -axis -axis ? ? ? ? ? ?

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? 𝒚 𝟐 𝒙 𝟐𝟎 ?

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

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)

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

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

Perpendicular Lines ! If two lines are perpendicular, then the gradient of one is the negative reciprocal of the other. Or: Gradient Gradient of Perpendicular Line ? -4 ? ? ? ? ?

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? ? Q2 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? Q3 Neither. Gradients are and . But , not -1, so not perpendicular. ?

Exercise 3 4 so perpendicular. 2 𝑦 3𝑥 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? ? 𝑙 5 1 A line goes through the indicated point and is 𝑥 Determine the equation of the line . ? 5 𝑦 𝑦 2 𝑥 4 𝑙 𝑥 ? Determine the equation of the line . Known point on : 3 So equation of : ? ?

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

Two last things Distance between two points Midpoint of two points ?5 ( 5,9 ) (3,6) 3 (𝟒,𝟕.𝟓) ? (3,6) Just find the average of and the average of . (7,9) 4 Find change and change to form right-angled triangle. Then use Pythagoras.

Past Exam Questions See GCSEPastPaper Solutions.pptx GCSERevision StraightLineEquations.docx