Modeling with tables, equations, and graphs (article) | Khan Academy
the x and y directions, the surface that represents a linear function in 3 dimensions will always Assume that the function f is represented by the following table. Determining Slopes from Equations, Graphs, and Tables . What is the slope of the linear function representing the relationship between the amount of time since leaving the Remember, slope can be found using the following formula. The equation of a line expresses a relationship between x and y values on the we can create a table of values for any line, these are just the x and y values.
This essential characteristic of linear relationships is called slope. The origin of that term is best related to the visual representation, where we can think of slope relating to the inclination of the line: For every 1-unit change horizontally, there is a 3-unit change vertically.
If we know two ordered pairs x1,y1 and x2,y2 that are part of a linear relationship, we have enough information to determine the slope of the relationship. The first line shown above represents the concept of slope: The second line shows how to calculate the slope when we know two ordered pairs in the relationship. The last symbol in the second line shows the traditional symbol used to represent slope, the letter m. Other important aspects of linear relationships include the location of the axes intercepts.
That is, we typically identify the ordered pairs that describe where the line intersects the x-axis and where it intersects the y-axis.
In the form of ordered pairs, that means we want to know the values a and b in the ordered pairs a,0 and 0,b. The first is the x-axis intercept and the second the y-axis intercept. When real-life situations are described through linear relationships, the intercepts often take on important meaning. The following example is intended to illustrate that.
Here is a table of values to show the cost associated with renting various numbers of Nintendo games. The cost includes the annual fee plus the per-night rental charge. We can also plot the ordered pairs number of rentals,total costas shown in the graph below. In symbolic form, we can write an equation to represent the relationship between games rented and total cost.
We have shown three different ways to represent the relationship: The fourth representation, verbal, was the original description of the relationship. How does the graph below differ from the one above?
How are they similar? What is the significance of the vertical-axis intercept?
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Why does only one end of the line segment have arrowheads on it? These and related questions are important to consider when working with representations of relationships.How to Determine if a Relationship Represented in a Table Is Linear & Write an Equation : Algebra
Here are some questions and answers that help focus on critical aspects of the relationship. Why or why not? How is this shown in the graph? Can a club member rent fewer than zero 0 games? The graph shows this because it stops at the ordered pair 0, In the real world, a member cannot rent a part of a game, because we assume each game is a physical whole that cannot be split into parts.
The graph is deceiving in this way, for the solid line seems to indicate that any value on the horizontal axis is possible. The output quantity is the dependent quantity. The value of the output depends on the value of the input. For each input, there is a single output. In the case of tossing a ball in the air, time is the input and height is the output.
Linear Equation Table
Remember the last time you were in a parking lot? Is this relation a function? Can you use the number of cars to correctly figure out the number of tires? Every single car has 4 tires, so the number of tires depends on how many cars are in the parking lot.
algebra precalculus - Graphing systems of linear equations. - Mathematics Stack Exchange
Every input of cars specifies a single possible output of tires. In this example, the relation of tires to cars is also a function—the number of tires also specifies the number of cars.
Now consider a different relation, between houses and the people who live in them. If an address is the input, and the output is the occupants, is this relation also a function? Think of your own house or apartment—are the people staying there always the same?
That time you went to camp, the occupancy changed. Every time you had a friend stay over, it changed again. Because a single address can produce more than one set of occupants, the relation is not a function. If you put the input in more than once, are you guaranteed to always get the same output?
With the cars and wheels, the answer is yes. For an input of 25 cars we always get an output of tires, no matter which 25 cars drive into that parking lot or when they arrive. The relation is a function. With the houses and occupants, the input of an address is not guaranteed to always produce the same output, because a house stays put while people come and go. The relation is not a function.