Category Archives: 2.2 Lines

2.2 Circles, 2.2 Lines, 2.3 Lines, Intercpets

X-intercepts and Y-intercepts

An x-intercept is where the graph touches or crosses the x-axis.

A y-intercept is where the graph touches of crosses the y-axis.

In this picture, the graph crosses the x-axis at the ordered pair (2, 0).  Since every ordered pair on the x-axis has a y coordinate of zero we can let y=0 to find x-intercepts.

To find an x-intercept: Let y=0 and solve for x.

In this picture, the graph crosses the y-axis at the ordered pair (0, 6).  Since every ordered pair on the y-axis has a x coordinate of zero we can let x=0 to find y-intercepts.

To find an y-intercept: Let x=0 and solve for y.

2.2 Lines, 2.3 Lines, Graphing

Graphing Linear Equations by Finding Intercepts

Steps for Graphing with the Intercept Method

  1. Find the x intercept and the y-intercept.
    • To find an x-intercept let y=0 and solve for x.
    • To find a y-intercept let x=0 and solve for y.
  2. Plot the x-intercept and y-intercept.
  3. Draw the line that connects the intercepts.

Example:   Graph the linear equation 2x-3y = 12

Solution:

 1. Find the x-intercept and the y-intercept.

To find an x-intercept: Let y=0 and solve for x.

2x-3y = 12

2x-3(0) = 12

2x-0 = 12

2x = 12

{2x}/2 = {12}/2

x = 6

The x-intercept of this equation is (6,0)

To find a y-intercept: Let x=0 and solve for y.

2x-3y = 12

2(0)-3y = 12

0-3y = 12

-3y = 12

{-3y}/{-3} = {12}/{-3}

y = -4

The y-intercept of this equation is (0,-4)

2. Plot the x-intercept and the y-intercept.

3. Draw the line that connects the intercepts.

 

Example:   Graph the linear equation y=3x

Solution:

 1. Find the x-intercept and the y-intercept.

To find an x-intercept: Let y=0 and solve for x.

y=3x

0=3x

{0}/3 = {3x}/3

0 = x

The x-intercept of this equation is (0,0)

To find a y-intercept: Let x=0 and solve for y.

y=3x

y=3(0)

y=0

The y-intercept of this equation is (0,0)

Since the x-intercept and the y-intercept are the same point and we need two distinct points to graph a line, we must find another ordered pair that is a solution to the equation.

Let x=1 and find the associated y value. (I chose x=1 but you could choose a different value)

y=3x

y=3(1)

y=3

Another ordered pair on the graph is (1,3)

2. Plot the x-intercept and the y-intercept.

3. Draw the line that connects the intercepts.

Example: Graphing a linear equation with intercepts.

Example: Graphing a linear equation with intercepts.

Example: Graphing a linear equation with intercepts.

2.2 Lines, 2.3 Lines, Slope

Calculating Slope

Given two points on the line   ( x_1, y_1) and   ( x_2, y_2), you can calculate the slope of a line by the following formula.

  m= {y_2-y_1} / {x_2-x_1}

  y_2-y_1 is also know as   Delta y or “the change in y.”

  x_2-x_1 is also know as   Delta x or “the change in x.”

 

  m= {y_2-y_1} / {x_2-x_1} ={Delta y} /{Delta x} ={rise}/{run} 

Example: Calculate the slope of the line containing the points   ( 5, 7) and   ( 9, 10).

Solution: Choose one of the points to be   ( x_1, y_1) and choose the other point to be   ( x_2, y_2).

I will choose   ( 5, 7)  to be   ( x_1, y_1)  and choose   ( 9, 10) to be   ( x_2, y_2).

Substitute these values into the slope formula and simplify.

  m= {y_2-y_1} / {x_2-x_1} ={10-7}/{9-5} =3/4

The slope of the line containing the points   ( 5, 7) and   ( 9, 10)  is m= 3/4.

Example: Calculate the slope of the line containing the points   ( -7, -2) and   ( 8, 8).

Solution: Choose one of the points to be   ( x_1, y_1) and choose the other point to be   ( x_2, y_2).

I will choose   ( -7, -2)  to be   ( x_1, y_1)  and choose   ( 8, 8) to be   ( x_2, y_2).

Substitute these values into the slope formula and simplify.

  m= {y_2-y_1} / {x_2-x_1} ={8-(-2)}/{8-(-7)}={8+2}/{8+7} =10/15=2/3

The slope of the line containing the points   ( -7, -2) and   ( 8, 8)  is m= 2/3.

Example: Finding the slope with the formula.

https://www.youtube.com/watch?v=1Cm7hjMUsrQ

Example: Finding the slope with the formula.

Example: Finding the slope from the graph.

 

2.2 Lines, 2.3 Lines, 4.1 Linear Functions and Their Properties, Slope

Interpretation of slope

The slope of a line is a number that indicates the “steepness” of a line.  Slope is usually denoted with the letter m.

If the slope of the line is positive, the line will be rising or increasing from left to right.

     

All three of the above graphs have a positive slope and the line is rising or increasing from left to right.  Notice as the value of the slope gets larger, the line is getting steeper.

If the slope of the line is negative, the line will be falling or decreasing from left to right.

   

All three of the above graphs have a negative slope and the line is falling or decreasing from left to right.  Notice as the value of the slope gets smaller, the line is getting steeper.

If the slope is zero, the line will be constant.  This results in a horizontal line.

     

All three of the the above graphs have a slope of zero.  The y values are constant.

A vertical line has a slope that is undefined.

   

All three of the vertical lines have undefined slope.