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Chapter 4: Problem 49

Determine whether the \(x\) -y values are generated by a linear function, a quadratic function, or neither. $$\begin{array}{rrrrrr} \hline x & 0 & 1 & 2 & 3 & 4 \\ y & -21 & -3 & 7 & 9 & 3 \\ \hline \end{array}$$

Short Answer

Expert verified
The values are generated by a quadratic function.

Step by step solution

01

Check for a Linear Pattern

To identify if the data set could be linear, calculate the difference between consecutive y-values (called the first differences). If these differences are constant, then the relationship is linear.\ \Calculate: \(-3 - (-21) = 18\) \(7 - (-3) = 10\) \(9 - 7 = 2\) \(3 - 9 = -6\) \Since the first differences \(18, 10, 2,\) and \(-6\) are not constant, the function is not linear.
02

Check for a Quadratic Pattern

For a quadratic function, the second differences should be constant when the first differences are not. Calculate the second differences by finding the differences of the first differences.\ \Calculate: \(10 - 18 = -8\) \(2 - 10 = -8\) \(-6 - 2 = -8\) \The second differences \(-8, -8, \) and \(-8\) are constant, indicating the function is quadratic.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Functions
Linear functions are like straight roads. They have a consistent pattern. For linear functions, when you look at the differences between consecutive y-values, known as the "first differences," these will always be the same. This means if you were to subtract each y-value from the next one, you would always get the same result.

When dealing with linear functions, here are some key aspects you should remember:
  • If the first differences are the same, the relationship between x and y is linear.
  • Linear equations are typically in the form of: \(y = mx + c\), where \(m\) is the slope, and \(c\) is the y-intercept.
In the given data, the first differences weren't constant. This led us to conclude that the function is not linear.
Patterns in Data
Patterns in data are the blueprint to understanding which type of function you are dealing with. By examining these regularities or distinctions, you can determine the type of relationship between variables. Recognizing the pattern helps you categorize the data.

Here's how you analyze it:
  • For linear functions, look for equal first differences.
  • For quadratic functions, the second differences will be equal.
  • If neither is constant, check for other types of functions, like exponential or cubic.
In our example, initially checking for a linear pattern didn't work as the first differences varied. It was only after checking the second differences that we noticed a consistent pattern, leading us to identify a quadratic function.
Second Differences
Second differences are an essential tool in identifying quadratic functions. Where the first differences look at consecutive y-values, second differences look at the differences of these first differences. This is akin to peeling another layer, giving us further insight into the relationship.

Here's what you need to know about second differences:
  • If the first differences are not consistent but the second differences are constant, the function is quadratic.
  • Quadratic functions often have the form: \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
In the example, the second differences were constant at -8, pointing out that the function is indeed quadratic. This consistency in the second differences helped confirm the nature of the function, solving the mystery of the initially irregular set of first differences.

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Most popular questions from this chapter

Sketch the graph of each rational function. Specify the intercepts and the asymptotes. (a) \(f(x)=(x-1)(x+2.75) /[(x+1)(x+3)]\) (b) \(g(x)=(x-1)(x+3.25) /[(x+1)(x+3)]\) [Compare the graphs you obtain in parts (a) and (b). Notice how a relatively small change in one of the constants can radically alter the graph.]

The functions \(f, g,\) and h are defined as follows: $$ f(x)=2 x-3 \quad g(x)=x^{2}+4 x+1 \quad h(x)=1-2 x^{2} $$ In each exercise, classify the function as linear, quadratic, or neither. $$g \circ f$$

Find the \(x\) -coordinate of the vertex of the parabola \(y=(x-a)(x-b) .\) (Your answer will be in terms of the constants \(a \text { and } b .)\) Hint: It's easier here to rely on symmetry than on completing the square.

(a) Complete the following table. Which \(x\) -y pair in the table yiclds the smallest sum \(x+y ?\) $$\begin{array}{llllllll} \hline x & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 \\ y & & & & & & \\ x y & 12 & 12 & 12 & 12 & 12 & 12 & 12 \\ x+y & & & & & & & \\ \hline \end{array}$$ (b) Find two positive numbers with a product of 12 and as small a sum as possible. Hint: The quantity that you need to minimize is \(x+(12 / x),\) where \(x>0 .\) But $$x+\frac{12}{x}=(\sqrt{x}-\sqrt{\frac{12}{x}})^{2}+2 \sqrt{12}$$ This last expression is minimized when the quantity within parentheses is zero. Why? (c) Use a calculator to verify that the two numbers obtained in part (b) produce a sum that is smaller than any of the sums obtained in part (a).

Sketch the graph of each rational function. Specify the intercepts and the asymptotes. $$y=2 x /(x+1)^{2}$$
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