- 1.2.1 Calculate the slope of a linear function and interpret its meaning.
- 1.2.2 Recognize the degree of a polynomial.
- 1.2.3 Find the roots of a quadratic polynomial.
- 1.2.4 Describe the graphs of basic odd and even polynomial functions.
- 1.2.5 Identify a rational function.
- 1.2.6 Describe the graphs of power and root functions.
- 1.2.7 Explain the difference between algebraic and transcendental functions.
- 1.2.8 Graph a piecewise-defined function.
- 1.2.9 Sketch the graph of a function that has been shifted, stretched, or reflected from its initial graph position.
We have studied the general characteristics of functions, so now let’s examine some specific classes of functions. We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define general algebraic functions and distinguish them from the transcendental functions we examine later in this chapter. We finish the section with examples of piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form.
Linear Functions and Slope
The easiest type of function to consider is a linear function. Linear functions have the form where and are constants. In Figure 1.15, we see examples of linear functions when is positive, negative, and zero. Note that if the graph of the line rises as increases. In other words, is increasing on If the graph of the line falls as increases. In this case, is decreasing on If the line is horizontal.
As suggested by Figure 1.15, the graph of any linear function is a line. One of the distinguishing features of a line is its slope. The slope is the change in for each unit change in The slope measures both the steepness and the direction of a line. If the slope is positive, the line points upward when moving from left to right. If the slope is negative, the line points downward when moving from left to right. If the slope is zero, the line is horizontal. To calculate the slope of a line, we need to determine the ratio of the change in versus the change in To do so, we choose any two points and on the line and calculate In Figure 1.16, we see this ratio is independent of the points chosen.
Consider line passing through points and Let and denote the changes in and respectively. The slope of the line is
We now examine the relationship between slope and the formula for a linear function. Consider the linear function given by the formula As discussed earlier, we know the graph of a linear function is given by a line. We can use our definition of slope to calculate the slope of this line. As shown, we can determine the slope by calculating for any points and on the line. Evaluating the function at we see that is a point on this line. Evaluating this function at we see that is also a point on this line. Therefore, the slope of this line is
We have shown that the coefficient is the slope of the line. We can conclude that the formula describes a line with slope Furthermore, because this line intersects the -axis at the point we see that the -intercept for this linear function is We conclude that the formula tells us the slope, and the -intercept, for this line. Since we often use the symbol to denote the slope of a line, we can write
to denote the slope-intercept form of a linear function.
Sometimes it is convenient to express a linear function in different ways. For example, suppose the graph of a linear function passes through the point and the slope of the line is Since any other point on the graph of must satisfy the equation
this linear function can be expressed by writing
We call this equation the point-slope equation for that linear function.
Since every nonvertical line is the graph of a linear function, the points on a nonvertical line can be described using the slope-intercept or point-slope equations. However, a vertical line does not represent the graph of a function and cannot be expressed in either of these forms. Instead, a vertical line is described by the equation for some constant Since neither the slope-intercept form nor the point-slope form allows for vertical lines, we use the notation
where are both not zero, to denote the standard form of a line.
Consider a line passing through the point with slope The equation
is the point-slope equation for that line.
Consider a line with slope and -intercept The equation
is an equation for that line in slope-intercept form.
The standard form of a line is given by the equation
where and are not both zero. This form is more general because it allows for a vertical line,
Finding the Slope and Equations of Lines
Consider the line passing through the points and as shown in Figure 1.17.
- Find the slope of the line.
- Find an equation for this linear function in point-slope form.
- Find an equation for this linear function in slope-intercept form.
- The slope of the line is
- To find an equation for the linear function in point-slope form, use the slope and choose any point on the line. If we choose the point we get the equation
- To find an equation for the linear function in slope-intercept form, solve the equation in part b. for When we do this, we get the equation
Consider the line passing through points and Find the slope of the line.
Find an equation of that line in point-slope form. Find an equation of that line in slope-intercept form.
A Linear Distance Function
Jessica leaves her house at 5:50 a.m. and goes for a 9-mile run. She returns to her house at 7:08 a.m. Answer the following questions, assuming Jessica runs at a constant pace.
- Describe the distance (in miles) Jessica runs as a linear function of her run time (in minutes).
- Sketch a graph of
- Interpret the meaning of the slope.
- At time Jessica is at her house, so At time minutes, Jessica has finished running mi, so The slope of the linear function is
The -intercept is so the equation for this linear function is
- To graph use the fact that the graph passes through the origin and has slope
- The slope describes the distance (in miles) Jessica runs per minute, or her average velocity.
A linear function is a special type of a more general class of functions: polynomials. A polynomial function is any function that can be written in the form
for some integer and constants where In the case when we allow for if the function is called the zero function. The value is called the degree of the polynomial; the constant is called the leading coefficient. A linear function of the form is a polynomial of degree 1 if and degree 0 if A polynomial of degree 0 is also called a constant function. A polynomial function of degree 2 is called a quadratic function. In particular, a quadratic function has the form where A polynomial function of degree is called a cubic function.
Some polynomial functions are power functions. A power function is any function of the form where and are any real numbers. The exponent in a power function can be any real number, but here we consider the case when the exponent is a positive integer. (We consider other cases later.) If the exponent is a positive integer, then is a polynomial. If is even, then is an even function because if is even. If is odd, then is an odd function because if is odd (Figure 1.18).
Behavior at Infinity
To determine the behavior of a function as the inputs approach infinity, we look at the values as the inputs, become larger. For some functions, the values of approach a finite number. For example, for the function the values become closer and closer to zero for all values of as they get larger and larger. For this function, we say approaches two as goes to infinity,” and we write as The line is a horizontal asymptote for the function because the graph of the function gets closer to the line as gets larger.
For other functions, the values may not approach a finite number but instead may become larger for all values of as they get larger. In that case, we say approaches infinity as approaches infinity,” and we write as For example, for the function the outputs become larger as the inputs get larger. We can conclude that the function approaches infinity as approaches infinity, and we write as The behavior as and the meaning of as or can be defined similarly. We can describe what happens to the values of as and as as the end behavior of the function.
To understand the end behavior for polynomial functions, we can focus on quadratic and cubic functions. The behavior for higher-degree polynomials can be analyzed similarly. Consider a quadratic function If the values as If the values as Since the graph of a quadratic function is a parabola, the parabola opens upward if the parabola opens downward if (See Figure 1.19(a).)
Now consider a cubic function If then as and as If then as and as As we can see from both of these graphs, the leading term of the polynomial determines the end behavior. (See Figure 1.19(b).)
Zeros of Polynomial Functions
Another characteristic of the graph of a polynomial function is where it intersects the -axis. To determine where a function intersects the -axis, we need to solve the equation for x. In the case of the linear function the -intercept is given by solving the equation In this case, we see that the -intercept is given by In the case of a quadratic function, finding the -intercept(s) requires finding the zeros of a quadratic equation: In some cases, it is easy to factor the polynomial to find the zeros. If not, we make use of the quadratic formula.
Consider the quadratic equation
where The solutions of this equation are given by the quadratic formula
If the discriminant this formula tells us there are two real numbers that satisfy the quadratic equation. If this formula tells us there is only one solution, and it is a real number. If no real numbers satisfy the quadratic equation.
In the case of higher-degree polynomials, it may be more complicated to determine where the graph intersects the -axis. In some instances, it is possible to find the -intercepts by factoring the polynomial to find its zeros. In other cases, it is impossible to calculate the exact values of the -intercepts. However, as we see later in the text, in cases such as this, we can use analytical tools to approximate (to a very high degree) where the -intercepts are located. Here we focus on the graphs of polynomials for which we can calculate their zeros explicitly.
Graphing Polynomial Functions
For the following functions a. and b., i. describe the behavior of as ii. find all zeros of and iii. sketch a graph of
- The function is a quadratic function.
- To find the zeros of use the quadratic formula. The zeros are
- To sketch the graph of use the information from your previous answers and combine it with the fact that the graph is a parabola opening downward.
- The function is a cubic function.
- Because As
- To find the zeros of we need to factor the polynomial. First, when we factor out of all the terms, we find
Then, when we factor the quadratic function we find
Therefore, the zeros of are
- Combining the results from parts i. and ii., draw a rough sketch of
Consider the quadratic function Find the zeros of Does the parabola open upward or downward?
A large variety of real-world situations can be described using mathematical models. A mathematical model is a method of simulating real-life situations with mathematical equations. Physicists, engineers, economists, and other researchers develop models by combining observation with quantitative data to develop equations, functions, graphs, and other mathematical tools to describe the behavior of various systems accurately. Models are useful because they help predict future outcomes. Examples of mathematical models include the study of population dynamics, investigations of weather patterns, and predictions of product sales.
As an example, let’s consider a mathematical model that a company could use to describe its revenue for the sale of a particular item. The amount of revenue a company receives for the sale of items sold at a price of dollars per item is described by the equation The company is interested in how the sales change as the price of the item changes. Suppose the data in Table 1.6 show the number of units a company sells as a function of the price per item.
In Figure 1.20, we see the graph the number of units sold (in thousands) as a function of price (in dollars). We note from the shape of the graph that the number of units sold is likely a linear function of price per item, and the data can be closely approximated by the linear function for where predicts the number of units sold in thousands. Using this linear function, the revenue (in thousands of dollars) can be estimated by the quadratic function
for In Example 1.15, we use this quadratic function to predict the amount of revenue the company receives depending on the price the company charges per item. Note that we cannot conclude definitively the actual number of units sold for values of for which no data are collected. However, given the other data values and the graph shown, it seems reasonable that the number of units sold (in thousands) if the price charged is dollars may be close to the values predicted by the linear function
A company is interested in predicting the amount of revenue it will receive depending on the price it charges for a particular item. Using the data from Table 1.6, the company arrives at the following quadratic function to model revenue (in thousands of dollars) as a function of price per item
- Predict the revenue if the company sells the item at a price of and
- Find the zeros of this function and interpret the meaning of the zeros.
- Sketch a graph of
- Use the graph to determine the value of that maximizes revenue. Find the maximum revenue.
- Evaluating the revenue function at and we can conclude that
- The zeros of this function can be found by solving the equation When we factor the quadratic expression, we get The solutions to this equation are given by For these values of the revenue is zero. When the revenue is zero because the company is giving away its merchandise for free. When the revenue is zero because the price is too high, and no one will buy any items.
- Knowing the fact that the function is quadratic, we also know the graph is a parabola. Since the leading coefficient is negative, the parabola opens downward. One property of parabolas is that they are symmetric about the axis, so since the zeros are at and the parabola must be symmetric about the line halfway between them, or
- The function is a parabola with zeros at and and it is symmetric about the line so the maximum revenue occurs at a price of per item. At that price, the revenue is
By allowing for quotients and fractional powers in polynomial functions, we create a larger class of functions. An algebraic function is one that involves addition, subtraction, multiplication, division, rational powers, and roots. Two types of algebraic functions are rational functions and root functions.
Just as rational numbers are quotients of integers, rational functions are quotients of polynomials. In particular, a rational function is any function of the form where and are polynomials. For example,
are rational functions. A root function is a power function of the form where is a positive integer greater than one. For example, is the square-root function and is the cube-root function. By allowing for compositions of root functions and rational functions, we can create other algebraic functions. For example, is an algebraic function.
Finding Domain and Range for Algebraic Functions
For each of the following functions, find the domain and range.
- It is not possible to divide by zero, so the domain is the set of real numbers such that To find the range, we need to find the values for which there exists a real number such that
When we multiply both sides of this equation by we see that must satisfy the equation
From this equation, we can see that must satisfy
If this equation has no solution. On the other hand, as long as
satisfies this equation. We can conclude that the range of is
- To find the domain of we need When we factor, we write This inequality holds if and only if both terms are positive or both terms are negative. For both terms to be positive, we need to find such that
These two inequalities reduce to and Therefore, the set must be part of the domain. For both terms to be negative, we need
These two inequalities also reduce to and There are no values of that satisfy both of these inequalities. Thus, we can conclude the domain of this function is
If then Therefore, and the range of is
Find the domain and range for the function
The root functions have defining characteristics depending on whether is odd or even. For all even integers the domain of is the interval For all odd integers the domain of is the set of all real numbers. Since for odd integers is an odd function if is odd. See the graphs of root functions for different values of in Figure 1.21.
Finding Domains for Algebraic Functions
For each of the following functions, determine the domain of the function.
- You cannot divide by zero, so the domain is the set of values such that Therefore, the domain is
- You need to determine the values of for which the denominator is zero. Since for all real numbers the denominator is never zero. Therefore, the domain is
- Since the square root of a negative number is not a real number, the domain is the set of values for which Therefore, the domain is
- The cube root is defined for all real numbers, so the domain is the interval
Find the domain for each of the following functions: and
Thus far, we have discussed algebraic functions. Some functions, however, cannot be described by basic algebraic operations. These functions are known as transcendental functions because they are said to “transcend,” or go beyond, algebra. The most common transcendental functions are trigonometric, exponential, and logarithmic functions. A trigonometric function relates the ratios of two sides of a right triangle. They are (We discuss trigonometric functions later in the chapter.) An exponential function is a function of the form where the base A logarithmic function is a function of the form for some constant where if and only if (We also discuss exponential and logarithmic functions later in the chapter.)
Classifying Algebraic and Transcendental Functions
Classify each of the following functions, a. through c., as algebraic or transcendental.
- Since this function involves basic algebraic operations only, it is an algebraic function.
- This function cannot be written as a formula that involves only basic algebraic operations, so it is transcendental. (Note that algebraic functions can only have powers that are rational numbers.)
- As in part b., this function cannot be written using a formula involving basic algebraic operations only; therefore, this function is transcendental.
Is an algebraic or a transcendental function?
Sometimes a function is defined by different formulas on different parts of its domain. A function with this property is known as a piecewise-defined function. The absolute value function is an example of a piecewise-defined function because the formula changes with the sign of
Other piecewise-defined functions may be represented by completely different formulas, depending on the part of the domain in which a point falls. To graph a piecewise-defined function, we graph each part of the function in its respective domain, on the same coordinate system. If the formula for a function is different for and we need to pay special attention to what happens at when we graph the function. Sometimes the graph needs to include an open or closed circle to indicate the value of the function at We examine this in the next example.
Graphing a Piecewise-Defined Function
Sketch a graph of the following piecewise-defined function:
Graph the linear function on the interval and graph the quadratic function on the interval Since the value of the function at is given by the formula we see that To indicate this on the graph, we draw a closed circle at the point The value of the function is given by for all but not at To indicate this on the graph, we draw an open circle at
Sketch a graph of the function
Parking Fees Described by a Piecewise-Defined Function
In a big city, drivers are charged variable rates for parking in a parking garage. They are charged $10 for the first hour or any part of the first hour and an additional $2 for each hour or part thereof up to a maximum of $30 for the day. The parking garage is open from 6 a.m. to 12 midnight.
- Write a piecewise-defined function that describes the cost to park in the parking garage as a function of hours parked
- Sketch a graph of this function
- Since the parking garage is open 18 hours each day, the domain for this function is The cost to park a car at this parking garage can be described piecewise by the function
- The graph of the function consists of several horizontal line segments.
The cost of mailing a letter is a function of the weight of the letter. Suppose the cost of mailing a letter is for the first ounce and for each additional ounce. Write a piecewise-defined function describing the cost as a function of the weight for where is measured in cents and is measured in ounces.
Transformations of Functions
We have seen several cases in which we have added, subtracted, or multiplied constants to form variations of simple functions. In the previous example, for instance, we subtracted 2 from the argument of the function to get the function This subtraction represents a shift of the function two units to the right. A shift, horizontally or vertically, is a type of transformation of a function. Other transformations include horizontal and vertical scalings, and reflections about the axes.
A vertical shift of a function occurs if we add or subtract the same constant to each output For the graph of is a shift of the graph of up units, whereas the graph of is a shift of the graph of down units. For example, the graph of the function is the graph of shifted up units; the graph of the function is the graph of shifted down units (Figure 1.23).
A horizontal shift of a function occurs if we add or subtract the same constant to each input For the graph of is a shift of the graph of to the left units; the graph of is a shift of the graph of to the right units. Why does the graph shift left when adding a constant and shift right when subtracting a constant? To answer this question, let’s look at an example.
Consider the function and evaluate this function at Since and the graph of is the graph of shifted left 3 units. Similarly, the graph of is the graph of shifted right units (Figure 1.24).
A vertical scaling of a graph occurs if we multiply all outputs of a function by the same positive constant. For the graph of the function is the graph of scaled vertically by a factor of If the values of the outputs for the function are larger than the values of the outputs for the function therefore, the graph has been stretched vertically. If then the outputs of the function are smaller, so the graph has been compressed. For example, the graph of the function is the graph of stretched vertically by a factor of 3, whereas the graph of is the graph of compressed vertically by a factor of (Figure 1.25).
The horizontal scaling of a function occurs if we multiply the inputs by the same positive constant. For the graph of the function is the graph of scaled horizontally by a factor of If the graph of is the graph of compressed horizontally. If the graph of is the graph of stretched horizontally. For example, consider the function and evaluate at Since the graph of is the graph of compressed horizontally. The graph of is a horizontal stretch of the graph of (Figure 1.26).
We have explored what happens to the graph of a function when we multiply by a constant to get a new function We have also discussed what happens to the graph of a function when we multiply the independent variable by to get a new function However, we have not addressed what happens to the graph of the function if the constant is negative. If we have a constant we can write c as a positive number multiplied by but, what kind of transformation do we get when we multiply the function or its argument by When we multiply all the outputs by we get a reflection about the -axis. When we multiply all inputs by we get a reflection about the -axis. For example, the graph of is the graph of reflected about the -axis. The graph of is the graph of reflected about the -axis (Figure 1.27).
If the graph of a function consists of more than one transformation of another graph, it is important to transform the graph in the correct order. Given a function the graph of the related function can be obtained from the graph of by performing the transformations in the following order.
- Horizontal scaling of the graph of by a factor of If reflect the graph about the -axis.
- Horizontal shift of the graph of If shift left. If shift right.
- Vertical scaling of the graph of by a factor of If reflect the graph about the -axis.
- Vertical shift of the graph of If shift up. If shift down.
We can summarize the different transformations and their related effects on the graph of a function in the following table.
|Transformation of||Effect on the graph of|
|Vertical shift up units|
|Vertical shift down units|
|Shift left by units|
|Shift right by units|
|Vertical stretch if
vertical compression if
|Horizontal stretch if horizontal compression if|
|Reflection about the -axis|
|Reflection about the -axis|
Transforming a Function
For each of the following functions, a. and b., sketch a graph by using a sequence of transformations of a well-known function.
- Starting with the graph of shift units to the left, reflect about the -axis, and then shift down 3 units.
- Starting with the graph of reflect about the -axis, stretch the graph vertically by a factor of 3, and move up 1 unit.
Describe how the function can be graphed using the graph of and a sequence of transformations.
Section 1.2 Exercises
For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical.
For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form.
Slope passes through
Slope passes through
Slope passes through
Passing through and
Passing through and
-intercept and -intercept
-intercept and -intercept
For the following exercises, for each linear equation, a. give the slope and -intercept b, if any, and b. graph the line.
For the following exercises, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the -intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither.
For the following exercises, use the graph of to graph each transformed function
For the following exercises, use the graph of to graph each transformed function
For the following exercises, use the graph of to graph each transformed function
For the following exercises, for each of the piecewise-defined functions, a. evaluate at the given values of the independent variable and b. sketch the graph.
For the following exercises, determine whether the statement is true or false. Explain why.
is a transcendental function.
is an odd root function
A logarithmic function is an algebraic function.
A function of the form where is a real valued constant, is an exponential function.
The domain of an even root function is all real numbers.
[T] A company purchases some computer equipment for $20,500. At the end of a 3-year period, the value of the equipment has decreased linearly to $12,300.
- Find a function that determines the value V of the equipment at the end of t years.
- Find and interpret the meaning of the - and -intercepts for this situation.
- What is the value of the equipment at the end of 5 years?
- When will the value of the equipment be $3000?
[T] Total online shopping during the Christmas holidays has increased dramatically during the past 5 years. In 2012 total online holiday sales were $42.3 billion, whereas in 2013 they were $48.1 billion.
- Find a linear function S that estimates the total online holiday sales in the year t.
- Interpret the slope of the graph of S.
- Use part a. to predict the year when online shopping during Christmas will reach $60 billion.
[T] A family bakery makes cupcakes and sells them at local outdoor festivals. For a music festival, there is a fixed cost of $125 to set up a cupcake stand. The owner estimates that it costs $0.75 to make each cupcake. The owner is interested in determining the total cost as a function of number of cupcakes made.
- Find a linear function that relates cost C to x, the number of cupcakes made.
- Find the cost to bake 160 cupcakes.
- If the owner sells the cupcakes for $1.50 apiece, how many cupcakes does she need to sell to start making profit? (Hint: Use the INTERSECTION function on a calculator to find this number.)
[T] A house purchased for $250,000 is expected to be worth twice its purchase price in 18 years.
- Find a linear function that models the price P of the house versus the number of years t since the original purchase.
- Interpret the slope of the graph of P.
- Find the price of the house 15 years from when it was originally purchased.
[T] A car was purchased for $26,000. The value of the car depreciates by $1500 per year.
- Find a linear function that models the value V of the car after t years.
- Find and interpret
[T] A condominium in an upscale part of the city was purchased for $432,000. In 35 years it is worth $60,500. Find the rate of depreciation.
[T] The total cost C to produce a certain item is modeled by the function where x is the number of items produced. Determine the cost to produce 175 items.
[T] A professor asks her class to report the amount of time t they spent writing two assignments. Most students report that it takes them about 45 minutes to type a four-page assignment and about 1.5 hours to type a nine-page assignment.
- Find the linear function that models this situation, where is the number of pages typed and t is the time in minutes.
- Use part a. to determine how many pages can be typed in 2 hours.
- Use part a. to determine how long it takes to type a 20-page assignment.
[T] The output (as a percent of total capacity) of nuclear power plants in the United States can be modeled by the function where t is time in years and corresponds to the beginning of 2000. Use the model to predict the percentage output in 2015.
[T] The admissions office at a public university estimates that 65% of the students offered admission to the class of 2019 will actually enroll.
- Find the linear function where is the number of students that actually enroll and is the number of all students offered admission to the class of 2019.
- If the university wants the 2019 freshman class size to be 1350, determine how many students should be admitted.