Digital Lesson on Graphs of Equations. Copyright © by Houghton Mifflin Company, Inc. All rights...

Digital Lesson on Graphs of Equations

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

The graph of an equation in two variables x and y is the set of all points (x, y) whose coordinates satisfy the equation.

For instance, the point (–1, 3) is on the graph of 2y – x = 7 because the equation is satisfied when –1 is substituted for x and 3 is substituted for y. That is,

2y – x = 7 Original Equation

2(3) – (–1) = 7 Substitute for x and y.

7 = 7 Equation is satisfied.


To sketch the graph of an equation,

1. Find several solution points of the equation by substituting various values for x and solving the equation for y.

2. Plot the points in the coordinate plane.

3. Connect the points using straight lines or smooth curves.


Example: Sketch the graph of y = –2x + 3.

1. Find several solution points of the equation.

x y = –2x + 3 (x, y)

–2 y = –2(–2) + 3 = 7 (–2, 7)

–1 y = –2(–1) + 3 = 5 (–1, 5)

0 y = –2(0) + 3 = 3 (0, 3)

1 y = –2(1) + 3 = 1 (1, 1)

2 y = –2(2) + 3 = –1 (2, –1)



2. Plot the points in the coordinate plane.

4 8

4

8

4

–4

x

yx y (x, y)

–2 7 (–2, 7)

–1 5 (–1, 5)

0 3 (0, 3)

1 1 (1, 1)

2 –1 (2, –1)



3. Connect the points with a straight line.

4 8

4

8

4

–4

x

y


Example: Sketch the graph of y = (x – 1)2.

x y (x, y)

–2 9 (–2, 9)

–1 4 (–1, 4)

0 1 (0, 1)

1 0 (1, 0)

2 1 (2, 1)

3 4 (3, 4)

4 9 (4, 9)

y

x2 4

2

6

8

–2


Example: Graph 122 xxy


Example: Sketch the graph of y = | x | + 1.

x y (x, y)

–2 3 (–2, 3)

–1 2 (–1, 2)

0 1 (0, 1)

1 2 (1, 2)

2 3 (2, 3)

y

x–2 2

2

4


The points at which the graph intersects the the x- or y-axis are called intercepts. A point at which the graph of an equation meets the y-axis is called a y-intercept. It is possible for a graph to have no intercepts, one intercept, or several intercepts.

If (x, 0) satisfies an equation, then the point (x, 0) is called an x-intercept of the graph of the equation.

If (0, y) satisfies an equation, then the point (0, y) is called a y-intercept of the graph of the equation.


To find the x-intercepts of the graph of an equation, substitute 0 for y in the equation and solve for x.

To find the y-intercepts of the graph of an equation algebraically, substitute 0 for x in the equation and solve for y.

Procedure for finding the x- and y- intercepts of the graph of an equation algebraically:


Example: Find the x- and y-intercepts of the graph of y = x2 + 4x – 5.

To find the x-intercepts, let y = 0 and solve for x.

0 = x2 + 4x – 5 Substitute 0 for y.

0 = (x – 1)(x + 5) Factor.

x – 1 = 0 x + 5 = 0 Set each factor equal to 0.

x = 1 x = –5 Solve for x.

So, the x-intercepts are (1, 0) and (–5, 0).

To find the y-intercept, let x = 0 and solve for y.

y = 02 + 4(0) – 5 = –5

So, the y-intercept is (0, –5).


To find the x-intercepts of the graph of an equation, locate the points at which the graph intersects the x-axis.

Procedure for finding the x- and y-intercepts of the graph of an equation graphically:

To find the y-intercepts of the graph of an equation, locate the points at which the graph intersects the y-axis.


Example: Find the x- and y-intercepts of the graph of x = | y | – 2 shown below.

y

x1

2

–3 2 3

The x-intercept is (–2, 0).

The y-intercepts are (0, 2) and (0, –2).

The graph intersects the x-axis at (–2, 0).

The graph intersects the y-axis at (0, 2) and at (0, –2).


Graphical Tests for Symmetry

• A graph is symmetric with respect to the y-axis if, whenever (x, y) is on the graph, (-x, y) is also on the graph. As an illustration of this we graph y = x2


Graphical Tests for Symmetry

• A graph is symmetric with respect to the x-axis if, whenever (x, y) is on the graph, (x, -y) is also on the graph. As an illustration of this we graph y2 = x.


Graphical Tests for Symmetry• A graph is symmetric with respect to the origin if,

whenever (x, y) is on the graph, (-x, -y) is also on the graph. As an illustration of this we graph y = x3


Algebraic Tests for Symmetry

• The graph of an equation is symmetric with respect to the y-axis if replacing x with –x yields an equivalent equation.

• The graph of an equation is symmetric with respect to the x-axis if

replacing y with –y yields an equivalent equation. • The graph of an equation is symmetric with respect to the origin if

replacing x with –x and replacing y with –y yields an equivalent equation.

The algebraic tests for symmetry are as follows:


Algebraic Tests for Symmetry

Example. The graph of y = x3 – x is symmetric with respect to the origin because:

xxy

xxy

xxy

xxy

3

3

3

3

)()(


Circles A circle with center at (h, k) and radius r consists of all points (x, y) whose distance from (h, k) is r. From the Distance Formula, we have the standard equation of a circle as:

222

22

)()(

)()(

rkyhx

rkyhx


Circles Example. Find the standard form of the equation of the

circle with center at (2, -5) and radius 4.

(x-2)2+(y-(-5))2=42

or (x-2)2+(y+5)2=16

Digital Lesson on Graphs of Equations. Copyright © by Houghton Mifflin Company, Inc. All rights...

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