Math-2. Lesson 3-1. Equations of Lines
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1 Math-2 Lesson 3-1 Equations of Lines
2 How can an equation make a line? y = x + 1 x Fill in the rest of the table rule x + 1 f(x) x Graph the x-y pairs + y - y + x
3 y = x + 1 So far we ve picked on integer values for x. We can also pick rational numbers between the integers. Fill in the rest of the table x y Graph the new x-y pairs on the same graph. + y x + x - y
4 y = x + 1 So far, we ve used integers and # s halfway in between. How many numbers are there between any 2 integers?. What would happen to our graph if we used every possible value of x as an input value into the function? More and more points are plotted. - x + y - y + x All of the points fall on the line.
5 Slope Intercept Form: An equation of the form y = mx + b. Where m = slope and b = y intercept. Y-intercept: The y-coordinate of a point where the graph intersects the y-axis. The x-coordinate of the y-intercept will always equal zero. (0, y) x-intercept: The x-coordinate of a point where the graph intersects the x-axis. The y-coordinate of the x-intercept will always equal zero. (x, 0)
6 1. What are the coordinates [(x, y) pair] of the x-intercept? 2. What are the coordinates [(x, y) pair] of the y-intercept? 3. What is the y-coordinate of the x-intercept? 4. What is the x-coordinate of the y-intercept?
7 Key Point (2, 0) x intercept y value is zero (0, -3) y intercept x value is zero
8 Standard form of a linear equation: An equation Of the form: Ax + By = C. Example: 3x + 4y = 12 Graphing Standard form equations x y Ordered Pair (0, 3) (4, 0) 1. Find the y-intercept (let x = 0) 3(0) + 4y = y = 12 y = 3 2. Find the x-intercept (let y = 0) 3x + 4(0) = 12 3x = 12 x = 4
9 Graph the lines. 2x + 3y = 6 4x - 3y = 12-4x + 2y = 8
10 Slope y = -2x 1 y = mx + b y-intercept: (0, -1) m 2 rise m run 2 1 m 2 rise m run 2 1 Run = 1 Rise = +2 Rise = -2 Run = +1
11 What is the equation of the line? y = -2x + 3 y = 3x 4 y = mx + b y-intercept: (0, -4) y = mx 4 m Rise = +3 Run = 1 rise run 3 1 m y = mx + b y-intercept: (0, 3) y = mx + 3 Rise = -2 Run = +1 rise run 2 1
12 Find the equation of a that passes through 2 points. (-2, 3) and (4, -3) Pick either point and substitute for x and y in the equation: (3) = (-1)(-2) + b Solve for b (the y-intercept) negative slope! Rise = -6 Run = 6 m rise run y = (-1)x + b (3) = 2 + b = b y = -x + 1
13 Find the equation of a that passes through 2 points. (3, -1) and (-2, 2) negative slope! Rise = -3 Run = 5 rise m run y x b Pick either point and substitute for x and y in the equation: 3 (3) ( 1) b 5 Solve for b (the y-intercept) b *5 * b b b b 12 y 3 x 5 5 Do not give the slope or y- intercept in decimal form.
14 What is the slope and y-intercept of the following equation? 2x 3y = 6 +3y + 3y 2x = 3y x 6 = 3y x 2 = y y = 2 3 x 2 Convert the following equation to slope intercept form Slope: 2/3 y-int: (0, -2)
15 Re-write slope intercept form as standard form Slope-intercept standard form y 2x 2-2x -2x 2x y 2 y 4 2( x 1) y 4 2x 2 y 2 x 1 3-2x -2x 2x y x y 2
16 What is the name of this form of equation? Slope intercept form y = mx + b y 2x 2 How can you recognize this form? (4 things) Y is a function of x (y is all by itself) There are no parentheses There are two variables The exponents of the variables are one.
17 What is the name of this form of equation? Standard form ax + by = c 2x 3y 6 How can you recognize this form? (4 things) Constant value is all by itself There are no parentheses There are two variables The exponents of the variables are one.
18 Parallel lines do not intersect each other. Which line is parallel to the line: y = 1? Is the red line parallel? Is the green line parallel? Is the yellow line parallel? Is the orange line parallel? Is the black line parallel? yes How can you tell if the graphed lines are parallel? Does the y-intercept help to make lines parallel? no
19 Parallel same slope y = mx + b Write the equation of a line that is parallel to the line y = 2x + 1 and passes through the point (0, 4) Slope = 2 y-intercept = 4 y = 2x + 4 Write the equation of a line that is parallel to the line y = 3x 4 and passes through the point (3, 8) Slope = 3 y-intercept =?? y = 3x + b (8) = 3(3) + b 8 = 9 + b b = -1 y = 3x 1
20 slopes of perpendicular lines are 1. reciprocals of each other. 2. opposite signs (+/-) of each other. Positive slope Is the slope positive or negative? Negative slope x 2 y 1 slope red 2 1 slope blue 1 2
21 What two things do we know about the slopes of perpendicular lines? The slopes are reciprocals of each other. The slopes have opposite signs of each other. The slopes of perpendicular lines are negative reciprocals of each other. What is the slope a line that is perpendicular to each of the following? y 2x 1 y 5 x 9 2 y 3 x 2 4 y 1 x 6 7
22 Find the slope intercept form of a line that is perpendicular to the line: y = 2x 6 and passes through the point (0, 1) Find the slope intercept form of a line that is perpendicular to the line: and passes through the point (5, 2) y 1 x 8 5
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