# Lesson 2.7

**Introduction**

**Course Objectives**

This lesson will address the following course outcomes:

· 21. Describe the behavior of common types of functions using words, algebraic symbols, graphs, and tables. Include descriptions of the dependent and independent variables.

**Specific Objectives**

Students will understand that

· that the family of the equation indicates the shape of the graph.

Students will be able to

· create a graph of an equation by plotting points.

In lesson 1.3 you learned that numbers and variables were used to create expressions, equations, and inequalities. In this lesson we will look more closely at equations, identify their families, and learn to graph them. We will begin with the fundamentals of graphing.

The first graph you used in this course was a number line. It is frequently used to show intervals of values resulting from solving inequalities that contain only one variable.

Graphs are more often used for showing the relationship between two variables. In such cases, two number lines are made to intersect each other at right angles to form a rectangular (or Cartesian) coordinate system. The horizontal axis is often called the x axis. The vertical axis is often called the y axis. To locate a point on the graph requires two numbers, the x coordinate and the y coordinate. These coordinates are written as an ordered pair (x,y). The two axes intersect at the origin (0,0).

To plot points, find the x coordinate on the x axis then move up or down until reaching the y coordinate. For example, to graph the ordered pair (2,3), find the 2 on the x axis, then move your pencil up 3 units.

Similarly, to plot (-3,-4), start at -3 on the x axis then go down 4 units.

#1 Points possible: 5. Total attempts: 5

Plot the points (-1,2), and (4,-2) on the graph.

Clear All Draw:

**Population Projections**

**Graphing Population Projections**

In Lesson 2.6 you found the absolute and relative change in the populations of several states. The data below is for Washington.

In 2000, the population in WA was 5,900,000.

In 2010, the population in WA was 6,700,000.

#2 Points possible: 5. Total attempts: 5

What was the absolute change in population for the decade?

#3 Points possible: 5. Total attempts: 5

What was the relative change in population for the decade? Round to 1 decimal place if needed.

%

Graphs, based on mathematical models, are often used for making projections into the future for planning purposes. People tend to believe official looking graphs, however one good critical thinking skill is to always question predictions about the future.

#4 Points possible: 12. Total attempts: 5

To open your thoughts to different possible futures for the population of Washington State, consider possible curves that include the two population values give above and extend out to 2020. Match each of the scenarios below to one of the graphs.

· The population of Washington continues to grow, but at a slower rate. People have stopped moving to the state.

· The population of Washington grows until there is a catastrophic event causing a large part of the population to flee the state

· The population of Washington continues to grow at a steady constant rate

· The population of Washington grows, and because it is such a great place to live there is a large influx of additional people who move to the state, driving up the population rapidly

a.

b.

c.

d.

#5 Points possible: 5. Total attempts: 5

In an earlier question, you calculated the absolute change in Washington’s population from 2000 to 2010. If the population continues to grow with the same absolute change, what will the population be in 2020?

people

#6 Points possible: 5. Total attempts: 5

In an earlier question, you calculated the relative change in Washington’s population from 2000 to 2010. If the population continues to grow with the same relative change, what will the population be in 2020?

people

**Families of Equations**

**Families of Equations**

The two graphs you just made will be used to help show you the difference between families of equations. The line that showed the same absolute change is a straight line. Straight lines are the type of line that results from a linear equation. The line that showed the same relative change is a curve that is typical of those produced from exponential equations. These are two of the three families that you should be able to recognize and graph.

The three families of equations that are explored in this course are linear equations, exponential equations and quadratic equations. Examples of these are shown in the table below.

Linear equation | 2x+3y=62x+3y=6 | y=12x−3y=12x-3 | y=3xy=3x |

Exponential equation | y=2xy=2x | y=(12)xy=(12)x | A=1000(1+0.03)tA=1000(1+0.03)t |

Quadratic equation | y=x2y=x2 | y=x2−4y=x2-4 | d=12gt2d=12gt2 |

#7 Points possible: 9. Total attempts: 5

Certain characteristics or features are unique to each family of equations. Looking at the examples in the table above, match each characteristic with the family it is unique to.

· x and y are first degree (not raised to a power)

· x is in the exponent

· x is squared (and y is not)

#8 Points possible: 9. Total attempts: 5

Identify each of the following equations as linear, exponential or quadratic

· y=3×2+6x+9y=3×2+6x+9

· y=50(1.04)xy=50(1.04)x

· y=−2x+5y=-2x+5

**Graphs of Families**

**Graphs of the Families of Equations**

Each family has a distinctive shape for its graph. Knowing the shape helps with graphing.

Graphs of linear equations produce straight lines.

Graphs of exponential equations produce J-shaped growth or decay curves.

Quadratic equations produce U-shaped parabolas.

One way to graph any equation is with a table of values. Before graphing, identify the family of the equation first, so you know the expected shape of the graph. Then use a table of values, by selecting x values, substituting them into the equation, and finding the y value. Plot the (x,y) ordered pairs and then connect the points with a line that extends to the borders of the grid.

To graph linear equations, select three x values. The first x value to be selected should be 0. To make your workload easier, the remaining 2 values you select should be numbers that cancel with the denominator of the fraction being multiplied times x. For example, for the linear equation y=12x−3y=12x-3 , select 0 then numbers such as 2, 4, 6, -2, -4. For the linear equation y=−53x+2y=-53x+2 , select 0 then numbers such as 3,6, -3, -6. By using numbers that can be divided by the denominator, your y value will not be a fraction, making it easier to graph.

Example: Graph y=12x−3y=12x-3

Table of values:

x | Substitution | Simplification | y | Ordered pair |

0 | y=12(0)−3y=12(0)-3 | y=0−3y=0-3 | -3 | (0, -3) |

2 | y=12(2)−3y=12(2)-3 | y=1−3y=1-3 | -2 | (2, -2) |

-2 | y=12(−2)−3y=12(-2)-3 | y=−1−3y=-1-3 | -4 | (-2, 4) |

In the next series of questions, to draw the graph, first select the appropriate tool below the graph, one of , , or . Click in the graph to place the first point, then click elsewhere to place the second point. You can move the points by clicking-and-dragging them. To remove a graph, drag a point outside the graph. If possible, use nice integer coordinates when you graph. For example, if you have a choice between plotting (-2, 4) and (1,23)(1,23) , use (-2, 4).

#9 Points possible: 21. Total attempts: 5

What is the family of the equation y=32x+4y=32x+4?

· linear

· quadratic

· exponential

Fill in the table with values for the equation.

X |
Y |

0 | |

2 | |

-2 | |

4 | |

-4 |

Graph the equation

Clear All Draw:

To graph exponential equations, keep in mind the expected shape. The first three x values to select should be 0, 1 and -1 because they will show if it is a growth or decay J curve. Select 0 because any value raised to the 0 power equals 1. Select 1 because any value raised to the first power equals itself. Select -1 because a value raised to the -1 power is the reciprocal. Plotting the ordered pairs for these three values should give you a reasonably good idea of what the graph will look like. Then you need to determine when it will go off the grid. It should go off the grid on one side just above the x axis and on the other side off the top of the grid.

Example: Graph y=2xy=2x

x | Substitution | Simplification | y | Ordered pair |

0 | y=20y=20 | 1 | (0, 1) | |

1 | y=21y=21 | 2 | (1, 2) | |

-1 | y=2−1y=2-1 | y=121y=121 | 1212 | (−1,12)(-1,12) |

-5 | y=2−5y=2-5 | y=125y=125 | 132132 | (−5,132)(-5,132) |

2 | y=22y=22 | 4 | (2,4)(2,4) | |

3 | y=23y=23 | 8 | Off the grid |

#10 Points possible: 21. Total attempts: 5

What is the family of the equation y=(4)xy=(4)x?

· linear

· quadratic

· exponential

Fill in the table with values for the equation. Enter *y* values as decimals rounded to 3 decimal places, not fractions.

X |
Y |

0 | |

1 | |

-1 | |

3 | |

-3 |

Graph the equation.

Clear All Draw:

To graph quadratic equations, plot each point as you calculate it. Keep in mind the expected shape of the parabola. Select x values that will help complete the shape. You almost always need to include some negative x values. Remember that squaring a negative number produces a positive number.

Example: Graph y=x2−3y=x2-3

x | Substitution | Simplification | y | Ordered pair |

0 | y=02−3y=02-3 | y=0−3y=0-3 | -3 | (0, -3) |

1 | y=12−3y=12-3 | y=1−3y=1-3 | -2 | (1, -2) |

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

2 | y=22−3y=22-3 | y=4−3y=4-3 | 1 | (2, 1) |

-2 | y=(−2)2−3y=(-2)2-3 | y=4−3y=4-3 | 1 | (-2, 1) |

3 | y=32−3y=32-3 | y=9−3y=9-3 | 6 | Off the grid |

#11 Points possible: 21. Total attempts: 5

What is the family of the equation y=−3×2+3y=-3×2+3?

· quadratic

· linear

· exponential

Fill in the table with values for the equation

X |
Y |

0 | |

1 | |

-1 | |

2 | |

-2 |

Graph the equation

**HW 2.7**

#1 Points possible: 5. Total attempts: 5

Match each graph with the corresponding function type.

·

·

·

#2 Points possible: 5. Total attempts: 5

Match each graph with the corresponding function type.

·

·

·

#3 Points possible: 5. Total attempts: 5

What is the family of the equation y=25x−4y=25x-4?

· exponential

· linear

· quadratic

Fill in the table with values for the equation.

X |
Y |

0 | |

5 | |

-5 | |

10 | |

-10 |

Graph the equation

Clear All Draw:

#4 Points possible: 5. Total attempts: 5

What is the family of the equation y=(3)xy=(3)x?

· exponential

· quadratic

· linear

Fill in the table with values for the equation. Enter *y* values as decimals rounded to 3 decimal places, not fractions.

X |
Y |

0 | |

1 | |

-1 | |

3 | |

-3 |

Graph the equation.

Clear All Draw:

#5 Points possible: 5. Total attempts: 5

What is the family of the equation y=−14x+4y=-14x+4?

· linear

· exponential

· quadratic

Fill in the table with values for the equation.

X |
Y |

0 | |

4 | |

-4 | |

8 | |

-8 |

Graph the equation

Clear All Draw:

#6 Points possible: 5. Total attempts: 5

What is the family of the equation y=−4×2+2y=-4×2+2?

· linear

· exponential

· quadratic

Fill in the table with values for the equation

X |
Y |

0 | |

1 | |

-1 | |

2 | |

-2 |

Graph the equation

Clear All Draw:

#7 Points possible: 5. Total attempts: 5

What is the family of the equation y=34x−2y=34x-2?

· linear

· exponential

· quadratic

Create a table of values for the equation on paper and use it to help you graph the equation.

Clear All Draw:

#8 Points possible: 5. Total attempts: 5

What is the family of the equation y=4×2−3y=4×2-3?

· quadratic

· linear

· exponential

Create a table of values for the equation on paper and use it to help you graph the equation.

Clear All Draw:

#9 Points possible: 5. Total attempts: 5

What is the family of the equation y=(12)xy=(12)x?

· linear

· quadratic

· exponential

Create a table of values for the equation on paper and use it to help you graph the equation.

Clear All Draw:

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