§ 1.3

Graphing Equations

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 1.3

The Rectangular Coordinate System

In the rectangular coordinate system, the horizontal number line is the x-axis.The vertical number line is the y-axis. The point of intersection of these axes istheir zero points, called the origin. The axes divide the plane into 4 quarters, called quadrants.

1st quadrant2nd quadrant

3rd quadrant 4th quadrant

x-axis

y-axis

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 1.3

The Rectangular Coordinate System

Each point in the rectangular coordinate system corresponds to an ordered pairof real numbers (x,y). Note the word “ordered” because order matters. The firstnumber in each pair, called the x-coordinate, denotes the distance and directionfrom the origin along the x-axis. The second number, the y-coordinate, denotes vertical distance and direction along a line parallel to the y-axis or along the y-axisItself.

In plotting points, we move across first (either left or right), andthen move either up or down, always starting from the origin.

Point Movement from origin

(3.5) 3 units right and 5 units up

(4,-3) 4 units right and 3 units down

(-2,-7) 2 units left and 7 units down

(0,0) 0 right or left and 0 up or down

Blitzer, Intermediate Algebra, 5e – Slide #4 Section 1.3

Plotting Points

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Plot the points (3,2) and (-2,-4).

(3,2)

(-2,-4)

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 1.3

The Graph of an Equation

The graph of an equation in two variables is the set of points whose coordinates satisfy the equation.

An ordered pair of real numbers (x,y) is said to satisfy the equation when substitution of the x and y coordinatesinto the equation makes it a true statement.

For example, in the equation

y = 2x + 6, the ordered pair (1,8) is a solution. When we substitute this pointthe sentence reads 8 = 8, which is true.

The ordered pair (2,3) is not a solution. When we substitute this point, the sentence reads 3 = 10, which is not true.

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 1.3

Graphing an Equation

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Graph y = 2|x| + 1.

x y = 2|x| + 1 Ordered Pair (x,y)-2 y = 2|-2| + 1 = 2(2) + 1 = 4 + 1 = 5 (-2,5)

-1 y = 2|-1| + 1 = 2(1) + 1 = 2 + 1 = 3 (-1,3)

0 y = 2|0| + 1 = 2(0) + 1 = 0 + 1 = 1 (0,1)

1 y = 2|1| + 1 = 2(1) + 1 = 2 + 1 = 3 (1,3)

2 y = 2|2| + 1 = 2(2) + 1 = 4 + 1 = 5 (2,5)

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 1.3

Graphing an Equation

CONTINUECONTINUEDD

(2,5)

(0,1)

(-1,3)

(-2,5)

(1,3)

In graphing an equation, wetry to get enough ordered pairsto get a good idea of what the graph looks like. Next,we plot these points. Finally, weconnect these points with a smooth curve or line, always moving from left to right. Thisoften gives us a picture of all ordered pairs that satisfy the equation.

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 1.3

Graphing an Equation

Graphs in the rectangular coordinate system can also be used to tell a story. Try to select the graph that best illustrates the story of the population of the U.S.A.

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Graph (c)

(a) (c)(b)

Years Years Years

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