2.3 Graph Equations of Lines

Parent function

The most basic function in a family of functions

y-intercept

The y coordinate of a point where the graph intersects the y - axis

Slope-intercept form

An equation of the form y= mx + b with slope m and y - intercept b

Standard form of a linear equation

The standard form of a linear equation is Ax + By= C where A and B are not both zero.

x-intercept

The coordinate of a point where a graph intersects the x – axis

PARENT FUNCTION FOR LINEAR FUNCTIONS

The parent function for the family of all linear functions is y = __x_. The graph of y = x is shown.

In general, a y- intercept of a graph is the y - coordinate of a point where the graph intersects the y-axis.

USING SLOPE-INTERCEPT FORM TO GRAPH AN EQUATION

Step 1 Write the equation in __slope-intercept__ form by solving for y.

Step 2 __Identify__ the y-intercept b and use it to plot the point (0, b) where the line crosses the y -axis.

Step 3 Identify the __slope__ m and use it to plot a second point on the line.

Step 4 __Draw__ a line through the two points.

Example 1

Graph an equation in slope-intercept form

Graph y = x + 1.

Step 1 The equation is already in slope-intercept form.

-3

Step 2 The y-intercept is __1__ , so plot the point (__0__,__1__) where the line crosses the__ y – axis__.

2

Step 3 The slope is _{______} or so plot a second point on the line by starting at (__0,1__) and then moving down _3_ units and right _2_ units. The second point is (__2,-2__).

Step 4 Draw a line through the two points.

USING STANDARD FORM TO GRAPH AN EQUATION

Step 1 Write the equation in standard form.

Step 2 Identify the x-intercept by letting __y_ = 0 and solving for __x__. Use the x-intercept to plot the point where the line crosses the x – axis.

Step 3 Identify the y-intercept by letting __x__ = 0 and solving for __y__. Use the y-intercept to plot the point where the line crosses the __y – axis__.

Step 4 Draw a line through the two points.

Example 2

Graph an equation in standard form

Graph 2x + 3y = 12.

Solution

Step 1 The equation is already in standard form.

Step 2

2x + 3(__0__) = 12 Let y = __0__

x = 6 Solve for x.

Plot the x-intercept at (__6__,0).

Step 3

2(__0__) + 3y = 12 Let x = __0__.

y = __4__ Solve for y.

Plot the y-intercept at (0,__4__).

Step 4 Draw a line through the two points.

HORIZONTAL AND VERTICAL LINES

Horizontal lines The graph of y = c is the horizontal line through (__0__,__c__).

Vertical lines The graph of x = c is the vertical line through (__c__,__0__).

Example 3

Graph horizontal and vertical lines

a. Graph y = -1 b. Graph x = 2.

Solution

a. The graph of y = -1 is the __horizontal__ line that passes through the point (0,_-1__). Notice that every point on the line has a y-coordinate of -1_.

b. The graph of x = 2 is the __vertical__ line that passes through the point (__2_,0). Notice that every point on the line has an x-coordinate of __2__.