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Functions and Their Graphs by J. Nicholas, J. Hunter, J.
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. The easiest type of function to consider is a linear function. One of the distinguishing features of a line is its slope. 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. The slope of the line is.
We now examine the relationship between slope and the formula for a linear function. 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. Therefore, the slope of this line is. Sometimes it is convenient to express a linear function in different ways. 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. Since neither the slope-intercept form nor the point-slope form allows for vertical lines, we use the notation. Definition: point-slope equation, point-slope equation and the standard form of a line.
The equation. To find an equation for the linear function in slope-intercept form, solve the equation in part b. When we do this, we get the equation. Jessica leaves her house at a. She returns to her house at a. Answer the following questions, assuming Jessica runs at a constant pace.
The slope of the linear function is. 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.
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. Some polynomial functions are power functions. 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. 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. If not, we make use of the quadratic formula. The solutions of this equation are given by the quadratic formula.
Here we focus on the graphs of polynomials for which we can calculate their zeros explicitly. The zeros are. Combining the results from parts i. 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. The company is interested in how the sales change as the price of the item changes. Using this linear function, the revenue in thousands of dollars can be estimated by the quadratic function. A company is interested in predicting the amount of revenue it will receive depending on the price it charges for a particular item.
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. 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. For example,. By allowing for compositions of root functions and rational functions, we can create other algebraic functions. This inequality holds if and only if both terms are positive or both terms are negative.
For both terms to be negative, we need. The denominator cannot be zero. Thus far, we have discussed algebraic functions. Some functions, however, cannot be described by basic algebraic operations.
The most common transcendental functions are trigonometric, exponential, and logarithmic functions. A trigonometric function relates the ratios of two sides of a right triangle. We also discuss exponential and logarithmic functions later in the chapter. Classify each of the following functions, a. 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. 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.
We examine this in the next example. In a big city, drivers are charged variable rates for parking in a parking garage. The parking garage is open from 6 a.
The cost to park a car at this parking garage can be described piecewise by the function. The cost of mailing a letter is a function of the weight of the letter. We have seen several cases in which we have added, subtracted, or multiplied constants to form variations of simple functions. 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.
Why does the graph shift left when adding a constant and shift right when subtracting a constant? 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. We can summarize the different transformations and their related effects on the graph of a function in the following table.
For each of the following functions, a. Learning Objectives Calculate the slope of a linear function and interpret its meaning. Recognize the degree of a polynomial. Find the roots of a quadratic polynomial. Describe the graphs of basic odd and even polynomial functions. Identify a rational function. Describe the graphs of power and root functions. Explain the difference between algebraic and transcendental functions. Graph a piecewise-defined function. Sketch the graph of a function that has been shifted, stretched, or reflected from its initial graph position.
Linear Functions and Slope The easiest type of function to consider is a linear function. 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.
Solution 1. Find an equation of that line in point-slope form.
This study focuses on connections between linear functions and their graphs that were made by tertiary remedial algebra students. The data consist of 63 responses to a written questionnaire and individual interviews with three participants. Furthermore, interpretations of the graph based on visual inspection appeared most useful when used in support of the algebraic approach. This is a preview of subscription content, access via your institution. Rent this article via DeepDyve. Arcavi, A. The role of visual representations in the learning of mathematics.
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terest, we consider the graphs of linear functions, quadratic functions, cubic To say that y is a function of x means that for each value of x there is exactly.
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We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials.
ReplyThe range is all real y ≥ −3. Example. Sketch x2 + y2 = 16 and explain why it is not the graph of a function. Solution x2 +.
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