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Chapter 5: Problem 8

Show that the following data can be modeled by a quadratic function, and find a formula for a quadratic model. $$ \begin{array}{|l|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline Q(x) & 5 & 6 & 13 & 26 & 45 \\ \hline \end{array} $$

Short Answer

Expert verified
The quadratic function is \( Q(x) = 3x^2 - 2x + 5 \).

Step by step solution

01

Determine the general equation form

A quadratic function can be expressed in the form \( Q(x) = ax^2 + bx + c \). Our task is to find the coefficients \(a\), \(b\), and \(c\) using the provided data points.
02

Establish equations using data points

Substitute each pair of \(x\) and \(Q(x)\) values into the quadratic formula: \( Q(x) = ax^2 + bx + c \).1. For \(x = 0\), \(Q(x) = 5\): \(c = 5\).2. For \(x = 1\), \(Q(x) = 6\): \(a(1)^2 + b(1) + c = 6\).3. For \(x = 2\), \(Q(x) = 13\): \(a(2)^2 + b(2) + c = 13\).This gives the system of equations:- \(a + b + 5 = 6\) \((1)\)- \(4a + 2b + 5 = 13\) \((2)\).
03

Simplify and solve for coefficients

First, simplify the equations derived:- From equation (1): \(a + b = 1\).- From equation (2): \(4a + 2b = 8\) which simplifies to \(2a + b = 4\) \((3)\).Now solve the system of equations:- From \(a + b = 1\), rearrange to \(b = 1 - a\).- Substitute \(b = 1 - a\) into \(2a + b = 4\): \[ 2a + (1 - a) = 4 \] \[ a + 1 = 4 \] \[ a = 3 \].
04

Determine remaining coefficients

Now substitute \(a = 3\) back into \(a + b = 1\): \[ 3 + b = 1 \] \[ b = -2 \]Recall \(c = 5\) from Step 2, equation for \(x=0\).
05

Write the quadratic equation

Using the values of \(a\), \(b\), and \(c\) we found, the quadratic model is given by: \[ Q(x) = 3x^2 - 2x + 5 \].

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Key Concepts

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

Quadratic Equation
A quadratic equation is a second-degree equation, which means it involves terms up to the power of two. It is generally written in the standard form as \( ax^2 + bx + c = 0 \). In this formula, \( a \), \( b \), and \( c \) are coefficients, where \( a \) cannot be zero, as this would make the equation linear instead of quadratic.

Understanding the structure of a quadratic equation helps us realize how the graph of the equation forms a clean curve called a parabola. A parabola can open upwards or downwards, depending on the sign of the coefficient \( a \).
  • If \( a > 0 \), the parabola opens upwards.
  • If \( a < 0 \), it opens downwards.
This basic shape and behavior are why quadratic equations are often used in modeling real-world phenomena where variables exhibit quadratic relationships.
Data Analysis
Data analysis in the context of quadratic functions involves examining the data to recognize patterns and establish a relationship between variables that is best represented by a quadratic model.

Let鈥檚 break it down further by taking a look at the data points provided: For \( x = 0, 1, 2, 3, \) and \( 4 \), the corresponding function values \( Q(x) \) are \( 5, 6, 13, 26, \) and \( 45 \) respectively.

By analyzing how the numbers change from one point to the next, and using differences between them, we can guide our creation of a quadratic model. It's essential to note that quadratic functions exhibit a uniform change in their second differences, a useful marker indicating that data can be modeled quadratically. This pattern is the key to determining whether a quadratic model fits a set of data.
Coefficients
Coefficients in a quadratic equation \( Q(x) = ax^2 + bx + c \) are the numerical factors \( a \), \( b \), and \( c \). Each coefficient serves a distinct role:

  • \( a \): Determines the direction and width of the parabola. A larger absolute value of \( a \) leads to a steeper parabola.
  • \( b \): Influences the slope of the parabola and the location of its vertex.
  • \( c \): Represents the y-intercept of the graph, informing us where the parabola crosses the y-axis.
Finding these coefficients from data involves forming equations from known data points and solving them systematically. By substituting data into the quadratic form and solving a system of equations, the values of \( a \), \( b \), and \( c \) can be determined, giving us the complete quadratic model.
Quadratic Model
A quadratic model is a mathematical representation that uses a quadratic function to approximate a set of data. Modeling a situation with a quadratic equation helps to predict outcomes and understand relationships.

For our data, after solving through step-by-step analysis, we arrived at the equation \( Q(x) = 3x^2 - 2x + 5 \). This model can now be utilized for interpolation within the given data range and potentially for extrapolation beyond it.

By employing such a model, the solutions can predict how future values might behave under the same conditions, a powerful tool in fields like physics, economics, and biology where nonlinear data modeling is frequent.

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Most popular questions from this chapter

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