9.8 Linear Functions

why do linear functions have this characteristic? sketch the reasoning 1) by definition.... 2) we can use this line.... 3) because the horizontal lines are all parallel 4) since they are right triangles, 5)thus, the lengths of the horizontal sides and the corresponding sides of these triangles represent 6) for a linear functions,

1) by definition, the graph of a linear function lies on a line. 2) we can use this line to form many different slope triangles which are right triangles that have a horizontal and a vertical side, and a side along the graph as shown in figure 9.69. 3) the angles they form with the graph of the function are all the same. 4) corresponding angles of these triangles must be equal; all such triangles are similar 5) changes in inputs because the horizontal axis represents inputs,; the lengths of vertical sides of these triangles represent the corresponding changes in outputs because the vertical axis represents outputs. 6) the change in input and the corresponding change in output are always in the same ratio

1) develop the equation for the arithmetic sequence 4 7 10 13 16 and indicate reasoning we can use to develop the equation 2) observe the important relationships between 3) each entry in the sequence is 4) the 0th entry for the sequence is the 5) the 0th entry shows .. 6) the constant increase of 3 in the arithmetic sequence tells us 7) constant rate of change is

1) y = 3x + 1 2) the way the arithmetic sequence grows, the 0th entry of the sequence and the equation and the graph for the sequence y = 3x + 1. 3) 3 more than the previous entry, and this 3 is the coefficient of x in the equation. \ 4) entry that would precede the first entry. so in this example it is 3 less than first entry 4. 5) where the line that the graph is on hits the y-axis and it appears as the constant term in the equation 6) that whenever the input increases by 1, the output increases by 3, and more generally, whenever the input increases, the output increase by 3 times as much. in other words, a change in input and the corresponding change in output are always in the same ratio, which in this case is 1 to 3 7) characteristics of linear functions in general and determines what kinds of real-world situations linear functions model

if two quantities vary together in a proportional relationship, in this way proportional relationships can be viewed as a function...

Then we can define variables for the quantities we can take one of the variables to be the independent variable and the other to be the dependent variable,

a key characteristic of linear functions is that

a change in inputs and the corresponding change in outputs are always in the same ratio, so that the rate of change of outputs per 1 unit of change in inputs is constant

every sequence can be viewed as

a function that has the counting numbers as inputs.

a linear function is

a function whose graph lies on a line in a coordinate plane

exponential functions

are functions that have an equation of the form y = a * b^x where a and b are positive numbers and b is not 1 generalize geometric sequences In the same way that linear functions generalize arithmetic sequences

we'll see that proportional relationships and arithmetic sequences

are linear functions

in chapter 7, we saw that if x and y are in an inversely proportional relationships, then

as x increases, y decreases. in contrast when x and y are in a proportional relationship, then as x increases, y must also increase. not all linear relationships behave this way. consider linear relationships whose graphs have a negative slope, as x increases, y decreases

to view a sequence as a function..

associate to the input x (a counting number), the output that is the xth entry in the sequence

remember that functions are relationships between values that vary together. thus to determine what kind of function a table exhibits...

consider how the independent and dependent variables change together.

suppose the tropical orchid company sells orchids by. mail order and charges $25 for each orchid plus 9$ shipping no matter how many orchids are ordered.

consider the associated function whose independent variable x is the number of orchids in an order and whose dependent variable y is the total cost to the customer for the order. this function is linear because for each additional orchid that is ordered, there is an additional cost of $25. in other words, because the cost increases at a steady rate of 25$ per orchid ordered

quadratic functions don't have a

constant rate of change the way that linear functions do, though they exhibit interesting pattern of change (figure 9.71) quadratic functions are characterized by a constant change in the change

linear relationships are characterized by

constant rate of change, so that the change in inputs and the change in outputs are always in the same ratio, and any change in output divided by the corresponding change in inputs gives the same constant rate

other kinds of relationships have

different characteristic patterns of change and these patterns of change determine the kinds of situations the functions model

how can you tell if a real-world relationship is modeled by a linear function?

examine how the outputs change in relation to a change in inputs and see if they are always in the same ratio. see if the outputs change at a steady rate relative to the inputs.

at a juice factory, juice concentrate and water must be mixed in a ratio of 3 to 5 to make a juice. . let x be a number of liters of concentrate and y a corresponding number of liters of water, so that for all values, x and y are in a ratio 3 to 5. represent this function using a table, strip diagram, and an equation for the proportional relationships how can we obtain the equation from the strip diagram how can we obtain an equation from the table what do you notice about the unit rate..

figure 9.66 represents this function multiple ways the 5 parts off water is 5/3 times as much as the the 3 parts of concentrate no matter how much is in each part, thus y = 5/3 * x first find the unit rate of 5/3 liters water or every 1 liter concentrate. for x liters concentrate, we use x times as much water, so the amount of water, y is 5/3 times x and thus y=5/3 * x. notice that the unit rate of water per 1 liter of concentrate is a key component off the equation

show a table and graph for the arithmetic sequence 4 7 10 13 16 the graph consists of isolated points because.. but we can see that the key points lie it turns out that

figure 9.67 isolated points because the function comes from a sequence on a line and so the function is linear arithmetic sequences (but not other kinds of sequences) are always linear functions

quadratic functions are

functions that have an equation of the form y = ax^2 + bx + c, where a b and c are numbers and a is not zero

what is the y-intercept

given a linear function that has independent variable x and dependent variable y, let b the be output that corresponds to the input 0 (which is assumed to be an allowable input) the number b is thus also the y-coordinate where the line crosses the y-axis and is called the y-intercept of the line

a proportional relationship always has a

graph on a line through the origin, thus proportional relationships are special kinds of linear functions

exponential functions like geometric sequences another patter of change is exponential functions model many situations in which

grow and decay in a characteristic way. like geometric sequences, whenever x increase by 1 unit y is multiplied by a fixed number figure 9.72. the change in an exponential function is again an exponential function, and it is proportional to the original function. because of this characteristic, exponential functions model many situations in which the growth of a quantity depends on how much of a quantity is present such as the amounts of money in a bank account or borrowed on credit, populations of people or animals, and the decay of radioactive substances

we'll see why straight line graphs

have a certain characteristic type of equation and how these equations relate to the graph and to real world situations

it is a linear function if...

if a function has the property that the change in inputs and the corresponding change in outputs are always in the same ratio

using the example above, in general

if quantities x and y vary together in a proportional relationship so that x and y are always in the ratio A to B, then x and y are related by an equation y = B/A * x

what is the slope

let me be the rate of change of y with respect to x, so that whenever x increases by 1, y increases by m. the number m is called the slope of the line and be found by dividing any change in y values by the corresponding change in x values

we sometimes refer to linear functions as

linear relationships

in figure 9.73, determinino the type of a function requires

looking at how x and y values change together. if we only look at the y-coordinates and ignore the x coordinates, we might think the table exhibits a linear this conclusion is not valid because because the x-coordintes in the table do not have a constant increase

many important real-world situations are

modeled by linear functions

equations for proportional relationships are

of a special type and are related to the ratio in a particular way

patterns of change are characteristic of characteristic of exponential functions include inversely proportional relationships vs linear relationships

quadratic functions geometric sequences graphs have a negative slope

what kinds of real world relationships are modeled by linear functions

the answer lies in the characteristics of liner functions

we have seen that for linear functions,

the change in inputs and the corresponding change in outputs are always in the same ratio. let's see how this characteristic leads to equations of the form y= mx + b for lines and linear functions

quadratic functions model

the height of an object that is falling (ignoring air resistance). according to physics, the acceleration due to gravity is constant. velocity is change in distance and acceleration is change in velocity, so situations of constant acceleration, such as falling objects, produce quadratic functions

where does the equation y = mx + b come from where are the corresponding y values?

think of going from 0 to a value of x on the x - axis. 2) when x is 0 y is b. as the input increases by x, the output increases by mx. so the y-value that corresponds with x is mx+ b. thus, y=mx + b is the equation for the line and the linear function

inversely proportional relationships are different from linear relationships whose graphs have a negative slope

we saw that in chapter 7 that for an inversely proportional relationship between x and y, the product x * y is always the same constant number. thus the relationship has the equation of the form x * y = c or y = c/x for some constant number c. not the equation of al linear function

constant of proportionality is

when working with an equation for a proportional relationships, the number B/A is often called a constant of proportionality this constant is a unit rate; it is the amount of the dependent variable per 1 unit of the independent variable