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CP Math Section 2-5 Point-Slope Form equation of a line. Quiz 2-3 1. Graph the line y = 2x + 1 2. Graph the line 3x + 4y = 12 3. Graph the line y = 2 4. Graph the line x = 3 2-5 Write Equations of Lines What we will learn: Finding the equation of a line given the following: 1. The slope and the y-intercept 2. A point on the line and the slope of the line. 3. 2 points on the line. 4. A point on the line and a parallel line. 5. A point on the line and a perpendicular line. Your turn: 1. What is the equation of this line? Your turn: 2. What is the equation of this line? Your turn: Name the correct form of the linear equation. 3. y 2x 2 4. 2x y 2 Slope intercept form Standard form y = mx + b Ax + By = C Two equations for the same line. How can you tell them apart? y 2x 2 2x y 2 ‘y’ is “alone”. The constant is “alone”. Two equations for the same line. Re-write one form as another form y 2x 2 -2x -2x 2x y 2 Slope-intercept standard form Get the constant alone on one side. Your turn. 5. Convert from: Slope-intercept to standard form 2 y x 1 3 6. Convert from: standard form to slope-intercept form 2 x y 1 Writing Equations of Lines What two things do you need in order to write the equation of a line in slope intercept form? y = mx + b Slope y-intercept The slope of a line is -3/4. The line crosses the y-axis at -5. What is the equation of the line? y = mx + b 3 y x5 4 Your turn. 7. The slope of a line is 2/3. It crosses the y-axis at -1. What is the equation of this line in slope-intercept form? 2 y x 1 3 8. The slope of a line is -3. It crosses the y-axis at 7. What is the equation of this line in slope-intercept form? y 3x 7 What information is needed? y = mx + b Slope and y-intercept Slope the slope equation. y y1 m x x1 Often times it is written like this: What property did we use to go from here… …to… y y1 m( x x1 ) …here? Point Slope Form of a Linear Equation y y1 m( x x1 ) Where ( x1 , y1 ) is a point on the line and ‘m’ is the slope of the line. Point Slope Form of a Linear Equation y y1 m( x x1 ) Used to find the equation of a line if: (1) Given a slope and a point on the line. (write on board) Finding the equation of a line: What is the equation of a line that has a slope of -2 and passes through the point (3, 4) ? y y1 m( x x1 ) y – 4 = -2(x – 3) convert to slope intercept form y – 4 = -2x + 6 y = -2x + 10 Finding the equation of a line: What is the equation of a line that has a slope of 3 and passes through the point (-1, -2) ? y y1 m( x x1 ) y – (-2) = 3(x – (-1)) convert to slope intercept form y + 2 = 3(x + 1) y + 2 = 3x + 3 y = 3x + 1 What information is needed? y mx b Slope and y-intercept y y1 m( x x1 ) Slope and a point Your turn: y y1 m( x x1 ) 9. What is the slope intercept equation of a line that passes through the point (-4, 2) and has a slope of -4 ? 10. What is the slope intercept equation of a line that passes through the point (3, -1) and has a slope of 6 ? Point Slope Form of a Linear Equation y y1 m( x x1 ) Used to find the equation of a line if: (1) Given a point on the line and the slope of the line. (write on board) Finding the equation of a line: What is the equation of a line that passes through the points (-3, 4) and (-2, 6) ? y y1 m( x x1 ) We need a point and the slope. Find the slope! y2 y1 64 m 2 (3) x2 x1 2 1 2 Plug slope and one of the points into the equation. y – 4 = 2(x – (-2)) y – 4 = 2(x + 2) y – 4 = 2x + 4 y = 2x + 8 Point Slope Form of a Linear Equation y y1 m( x x1 ) Used to find the equation of a line if: (1) Given a slope and a point on the line. (2) Given 2 points on the line. (write on board) Your turn: If you were given the following information, what form of equation (slope-int. or point-slope) would you use to come up with the specific equation of the line? 11. Slope = 3, crosses y-axis at (0, -2) y mx b 12. (slope-intercept form) Slope = -2, passes thru the point (1, 8) y y1 m( x x1 ) (point – slope form) Your turn: 13. What is the equation of a line that passes through points (-1, 5) and (1, -5) ? 14. What is the equation of a line that passes through points (0, 7) and (3, 5) ? Parallel and Perpendicular Lines Parallel lines have the same _______. Perpendicular lines have ______ that are negative ___________ of each other. Your turn: 15. What is the slope of the line? y 2x 3 16. Are the following lines parallel, perpendicular or neither? y 2x 3 y 2 x 5 17. Are the following lines parallel, perpendicular or neither? 2 3 y 3 x4 y x6 2 18. Are the following lines parallel, perpendicular or neither? 5 4 3 1 y x 5 5 y 3 x 5 Equations of Parallel Lines How do the slopes of parallel lines compare? Parallel same slope Find the equation of a line that is parallel to the line y = 2x + 1 and passes through the point (4, -1) What equation form will you use? y y1 m( x x1 ) What is the slope of the other line? m2 What is the slope of the our line? m2 Plug the slope and a point into the equation. y (1) 2( x 4) Equations of Parallel Lines Parallel same slope Find the equation of a line that is parallel to the line y = 2x + 1 and passes through the point (4, -1) y y1 m( x x1 ) y (1) 2( x 4) y 1 2x 8 y 2x 9 Convert to slope intercept form. Point Slope Form of a Linear Equation y y1 m( x x1 ) Used to find the equation of a line if: (1) Given a slope and a point on the line. (2) Given 2 points on the line. (3) Given a parallel line and a point. (write on board) Find the equation of the line Find the equation of a line that is perpendicular to the line y = 2x – 6 and passes through the point (0, 1) What equation form will you use? What is the slope of the other line? What is the slope of the our line? y y1 m( x x1 ) m2 m1 Plug the slope and a point into the equation. y 1 1 ( x 0) 2 2 Find the equation of the line Find the slope intercept form of a line that is perpendicular to the line: y = 2x – 6 and passes through the point (0, 1) y y1 m( x x1 ) y 1 1 ( x 0) 2 y 1 1 x 2 Convert to slope intercept form. y 1 x 1 2 Point Slope Form of a Linear Equation y y1 m( x x1 ) Used to find the equation of a line if: (1) Given a slope and a point on the line. (2) Given 2 points on the line. (3) Given a parallel line and a point. (4) Given a perpendicular line and a point. (write on board) Your Turn: 19. Find the slope intercept form of a line that is perpendicular to the line: y = -1/3 x – 2 and passes through the point (4, 5) Three equations for the same line. Re-write one form as another form y 4 2( x 1) Point slope standard form Get the “x’s and “y’s” alone on one side. y 4 2 x 2 (distributive property) -2x -2x 2 x y 4 2 +4 +4 2x y 2 Your turn: 20. Convert from Point slope to standard form y 2 2 ( x 3) 3 21. Given the “point-slope” form of a linear equation, rewrite the equation in “slope intercept” form. y 2 3( x 1) Fractions!!! 1 2 5 1 2 5 1 1 10 5 5 9 5 1 2 5 * 5 1 5 Your turn: 22. 2 4 3 23. 3 2 7 Simplify the following: Point Slope Form of a Linear Equation Often times fractions are introduced into these equations. What is the equation of a line that passes through points (2, -3) and (-1, 7) ? y y1 m( x x1 ) slope. 7 (3) 10 10 m 1 2 3 3 10 y 7 ( x (1)) 3 10 y 7 ( x 1) 3 Point Slope Form of a Linear Equation Often times fractions are introduced into these equations. What is the equation of a line that passes through points (2, -3) and (-1, 7) ? Step 3: Plug a point and the slope into the equation. 10 y 7 ( x 1) 3 10 10 y7 x 3 3 10 10 y x 7 3 3 10 10 7 y x 3 3 1 10 10 7 3 y x * 3 3 1 3 10 10 21 y x 3 3 3 10 11 y x 3 3 Your turn: 24. What is the equation of a line that passes through the points (-2, 3) and (3, -5) ? Equations of Parallel Lines Parallel same slope parallel to the line y = 2x + 1 and passes through the point (4, -1) Slope = 2 ( x1, y1 ) (4,1) y y1 m( x x1 ) y (1) 2( x 4) y 1 2x 8 y 2x 9 Your turn: 25. What is the equation of a line that passes through the point (2, -7) and is parallel to the line: y = -x - 3