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Chapter 2: Problem 29

analyze the differences in the outputs to determine whether the data are linear, quadratic, or neither. Explain. If linear or quadratic, write an equation that fi ts the data. $$ \begin{array}{|l|c|c|c|c|c|} \hline \begin{array}{l} \text { Price decrease } \\ \text { (dollars), } \boldsymbol{x} \end{array} & 0 & 5 & 10 & 15 & 20 \\ \hline \begin{array}{l} \text { Revenue } \\ \mathbf{( \$ 1 0 0 0 s ) ,} \boldsymbol{y} \end{array} & 470 & 630 & 690 & 650 & 510 \\ \hline \end{array} $$

Short Answer

Expert verified
The relationship between price decrease and revenue is neither linear nor quadratic as the differences between the y-values are not consistent.

Step by step solution

01

Determine the Type of Relationship

Start by plotting the given points on a graph. Based on the shape of the graph, you can usually tell if the relationship is linear (straight line), quadratic (parabola), or neither.
02

Calculate the Differences between y-values

If there is uncertainty, calculate the differences between the y-values. A consistent difference indicates a linear relationship, a consistent difference of differences indicates a quadratic relationship. In our case, the differences are not consistent, implying a non-linear relationship.
03

Verify the type of relationship

The graph should resemble a parabolic curve pointing downwards, which further suggests a quadratic relationship. However, given the inconsistent differences in step 2, it is clear that the relationship is neither linear nor quadratic.

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Key Concepts

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

Linear Relationship
A linear relationship occurs when two variables have a constant rate of change. This means that when you plot the data on a graph, it forms a straight line. To determine if a relationship is linear, you can calculate the differences between consecutive y-values and see if these differences are consistent across the dataset. If they are, the data likely follows a linear pattern.
  • Straight line on graph
  • Consistent first differences
In the provided example, the revenue does not show a consistent difference in y-values, thus indicating that the relationship between price decrease and revenue is not linear.
Quadratic Relationship
A quadratic relationship is present when data points form a parabolic shape, usually opening upwards or downwards on a graph. This type of relationship suggests that the change between y-values is not constant, but the difference of these changes (or second difference) remains consistent.
  • Parabolic curve on graph
  • Consistent second differences
The data in the example fails to show consistent second differences, leading us to conclude that the relationship is neither quadratic.
Parabolic Curve
A parabolic curve is a type of graph produced by a quadratic function. It can look like a U-shape or an inverted U. The key characteristic of a parabolic curve is that it exhibits a symmetrical pattern about a vertical axis, called the axis of symmetry.
  • Symmetrical U-shape or inverted U-shape
  • Axis of symmetry
In our exercise, while the graph of revenue vs. price decrease might loosely resemble a downward opening parabola, the lack of consistent quadratic differences means we cannot confirm a true parabolic curve as dictated by a quadratic relationship.
Difference of Differences
The concept of difference of differences helps identify a quadratic relationship. This involves taking the difference between consecutive y-values to get the first differences, then calculating the differences between these first differences to find the second differences.
  • First differences: changes between consecutive y-values
  • Second differences: differences between first differences
When the second differences are consistent, you might have a quadratic relationship; however, in our scenario, the second differences vary, confirming the absence of a quadratic relationship.

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Most popular questions from this chapter

OPEN-ENDED Write two different quadratic functior in intercept form whose graphs have the axis of symmetry \(x=3\).

The \(x\)-coordinate of the vertex is $$ x=\frac{b}{2 a}=\frac{24}{2(4)}=3 $$

\(y=-4(x-2)^2+4\)

WRITING Two quadratic functions have graphs with vertices (2, 4) and (2, ?3). Explain why you can not use the axes of symmetry to distinguish between the two functions.

\(f(x)=8 x^2-6\); horizontal stretch by a factor of 2 and a translation 2 units up, followed by a reflection in the \(y\)-axis
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