Section 2.3 Linear Functions and Slopes

The Slope of a Line

Example Find the slope of the line passing through the pair of points. (5,-2) and (-1,7)

The Point-Slope Form of the Equation of a Line

Solving in both forms A.Write the equation in point slope form of the line with slope 4 that passes through the point (4,-3). B.Then solve the equation for y y1y1 x1x1 y-(-3) = 4(x-4) Substituting the values into the euation y+3 = 4(x-4) This is Point Slope Form. Apply the distributive property for the parentheses. This will give us the slope intercept form. (The equation is solved for y.) y= 4(x-4)-3 y= 4x-16-3 y-y 1 = m(x-x 1 ) (slope intercept form) Y=4x-19

Example Write the point slope form of the equation of the line with slope of -4 that passes through (2,5). Then solve for y.

If you are given two points and you need to write an equation in point-slope form, then you can use either point for (x 1,y 1 ).

Example Write the point slope form of the equation of the line that passes through (2,5) and (-1,0). Then solve for y.

The Slope-Intercept Form of the Equation of a Line

Point Slope Form For a nonvertical line with slope m that passes through (x 1,y 1 ) the equation is y-y 1 = m(x-x 1 ) Example: slope = -3 point on the line(-1,-2) Y-(-2)= -3(x-(-1)) Y+2= -3(x+1) Slope Intercept Form For a nonvertical line with slope m and y- intercept b the equation is y=mx+b Example: slope =2 y-intercept of 6 Y=2x +6 Two forms for Equations of Lines

Graph the linear equation y= 2/3x+4 First: Plot the y-intercept of 4 Rise by 2 units Run ( go to the right) by 3 units. Plot the second point (3, 6) Connect the two points with a straight edge or ruler. (0,4) (3,6)

Example Graph the linear equation y= -3x+5

Example

Equations of Horizontal and Vertical Lines

Example Graph x=4. Graph y=-2

The General Form of the Equation of a Line

Y intercept slope

Example Find the slope and the y intercept of the line whose equation is 2x+5y-10=0.

Using Intercepts to Graph Ax + By + C = 0

Find x and y intercepts to graph a line 6x-2y=12 X intercept so let y=0Y intercept so let x= x-2(0)=12 6x=12 X=2 (2,0) Y=-6 (0,-6) -2y=12 6(0)-2y=12

Example Find the x and y intercepts then graph using those points. X-4y-8=0

Summary

Applications

Example Diameter Price

Graphing Calculator-Linear Regression D$ More on the next slide.

Graphing Calculator-Linear Regression continued

(a) (b) (c) (d) Find the equation of the line in slope- intercept form for a line that passes through (0,-4) and has a slope of -2.

(a) (b) (c) (d) Find the equation of the line in slope- intercept form of the line that passes through (-3,-2) and (0,-2).

(a) (b) (c) (d) What is the slope of the line 3x - 7y – 4 = 0.