Warm-up: Check the equation y = 3x – x3 for symmetry. Homework: Pg. 128 (11, 13, 15, 23, 25, 35, 39, 41, 45, 49, 51, 55, 66, 67, 71, 73, 75) Write all equations in standard form.

Objective The student will be able to: Find the slope of a line given 2 points and a graph. Graph a line given its equation. Find the equation of a line given two points. Find the equations of parallel and perpendicular lines. Transform equations into standard form

Slope is the steepness of a line. What does the 7% mean? 7% 7% is the slope of the road. It means the road drops 7 feet vertically for every 100 feet horizontally. 7 feet 100 feet Slope is the steepness of a line.

Slope can be expressed different ways:

Determine the slope of the line. rise 3 = = run 6 6 3 Slope is positive so the line increases!

Finding Slopes of Horizontal and Vertical Lines Find the slope of each line. A. B.   You cannot divide by 0   The slope is undefined. The slope is 0.

Slope

Find the slope of the line that passes through the points (-2, -2) and (4, 1).

Graphing Linear Equations The formula for Slope-Intercept Form is: b is the y-intercept. m is the slope.

2) Use rise and run of the slope to plot more points 1) Plot the y-intercept run 1 2) Use rise and run of the slope to plot more points rise 2 run 1 rise 2  

Sometimes we re-write the equation in slope-intercept form to graph. The constant, b = 3 is the y-intercept. The coefficient, m = -2 is the slope.

Point-Slope Form If we’re given either: One point on the line that is not the y-intercept and the slope Two points on the line and neither are the y-intercept Then: We use point-slope form y – y1 = m(x – x1) where m = slope and (x1, y1) is one of the points on the line.

Using a Point and a Slope to Write an Equation Write an equation in slope-intercept form for the line passing through (1, -2) with slope 4. Step 1 Substitute the slope and the point into the point-slope form. y – y1 = m(x – x1) y – (–2) = 4(x – 1) Step 2 Write the equation in slope-intercept form. y + 2 = 4(x – 1) y + 2 = 4x – 4 y = 4x – 6

Using Two Points to Write an Equation Write an equation in slope-intercept form for the line through the two points. (2, –3) and (4, 1) Step 1 Find the slope. Step 2 Substitute the slope and one of the points into the point-slope form. y – y1 = m(x – x1) Choose (2, –3). y – (–3) = 2(x – 2) Step 3 Write the equation in slope-intercept form. y + 3 = 2(x – 2) y = 2x – 7 y + 3 = 2x – 4

Transforming Equations into Standard Form: ax + by = c How? 1) Use properties of equality to get x-term, y-term and constant in proper places. 2) Remove Fractions by multiplying. If necessary, multiply by a negative to make the leading coefficient positive.

Example: Write the equation in standard form.            

Equations of lines to Remember Slope-Intercept Form Useful for graphing Point-Slope Form Use this form when you know a point on the line and the slope if you have two points on the line first find the slope and then use one of the points and the slope in this equation. Standard Form Vertical Line Horizontal Line

Parallel and Perpendicular Parallel lines have the same slopes m1 = m2 Perpendicular lines have opposite reciprocal slopes  

4 -1 2 y = - x + 1 1 Let's look at a line and a point not on the line Find the equation of a line parallel to y = - x + 1 that passes through the point (2, 4) y = - x + 1 1 y = - x + 1 (2, 4) m = –1 4 -1 2 So we know the slope is –1 and it passes through (2, 4). Distribute and then solve for y to leave in slope-intercept form.

4 2 y = - 2x + 1 What if we wanted perpendicular instead of parallel? Find the equation of a line perpendicular to y = - 2x + 1 that passes through the point (2, 4) y = - 2x + 1 The slope of the first line is –2 (2, 4) The slope of the line perpendicular to the first line is found by taking the opposite reciprocal   4 2 Distribute and then solve for y to put in slope-intercept form. So the slope of a perpendicular line is 1/2 and it passes through (2, 4).

Summary: The student will be able to: Find the slope of a line given 2 points and a graph. Graph a line given its equation. Find the equation of a line given two points. Find the equations of parallel and perpendicular lines. Transform equations into standard form

Sneedlegrit: Find the equation of the line perpendicular to y = 3x – 2 that contains the point (-1, 4) Homework: Pg. 128 (11, 13, 15, 23, 25, 35, 39, 41, 45, 49, 51, 55, 66, 67, 71, 73, 75) Write all equations in standard form.