Question

The table shows the number of tiles in each fi gure. Verify that the data show a quadratic relationship. Predict the number of tiles in the 12th figure. cant copy graph $$ \begin{array}{|l|c|c|c|c|} \hline \text { Figure } & 1 & 2 & 3 & 4 \\ \hline \text { Number of Tiles } & 1 & 5 & 11 & 19 \\ \hline \end{array} $$

Short answer

The relationship between the Figure number and the Number of Tiles is quadratic. The quadratic equation representing this relationship is \( y=x^2+x-1 \). The 12th figure will have 155 tiles.

Step-by-step solution

Understand the Quadratic Relationship

A quadratic relationship between two variables x and y is represented as \( y=ax^2+bx+c \), where a, b and c are constants. For a pattern of numbers to show a quadratic relationship, the second difference (difference of difference between consecutive numbers) should be constant. Let's test this first.

Calculate Consecutive Differences

This involves identifying the difference between each consecutive pairs of 'Number of Tiles'. Difference between 5 and 1 is 4, between 11 and 5 is 6 and between 19 and 11 is 8.

Calculate Second Differences

This step involves calculating the difference between each consecutive pairs of first differences obtained in Step 2 are: Difference between 6 and 4 is 2 and between 8 and 6 is 2.

Verify Quadratic Relationship

The second differences are constant (both are 2), which means the Number of Tiles shows a quadratic relationship with the Figure number.

Formulate Equation

The quadratic equation has the form \( y=ax^2+bx+c \). Since the second difference is 2, which is 2a, a equals 1. To find b and c, substitute the values of x and y from table into the equation to create a system of linear equations and solve it. Doing so, it is found that b=1 and c=-1.

Predict the Number of Tiles

Substitute 12 (the 12th figure) into the equation \( y=x^2+x-1 \) to get the predicted number of tiles for the 12th figure. Thus \( y = 12^2 + 12 -1 = 155 \), hence 155 tiles are needed for the 12th figure.

Key concepts

Understanding Second Differences in Quadratic Relationships

Second differences are a helpful tool for identifying quadratic relationships in sequences of numbers. First, let's delve into what these mean. In any numerical sequence, we might begin by calculating the first difference: subtract each term from the next one. For instance, in our exercise, the sequence of tile numbers starts as 1, 5, 11, 19. Calculating their differences gives us 4, 6, and 8. If we find the difference between these first differences and call this the "second difference," they help us diagnose the nature of the relationship. For this data, the second differences are consistent at 2, calculated by subtracting 4 from 6 and 6 from 8. A constant second difference shows us that the sequence forms a quadratic pattern. If these differ consistently, the pattern is linear.

• First difference: Differences between successive terms.
• Second difference: Differences between successive first differences.
• Constant second difference indicates a quadratic relationship.

Exploring Quadratic Equations as General Patterns

Quadratic equations are the mathematical representation of quadratic relationships. They use the general formula, \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. This type of equation indicates that the relationship between the sequences' terms and their positions is parabolically natured and nonlinear. In our given exercise, we discovered a quadratic relationship by respecting the second differences. From there, we sought to form a specific quadratic equation. After realizing that the second difference was 2, we derived \( a \) to be 1 using the relation \( 2a = 2 \). Subsequently, by substituting the values for figures and their corresponding number of tiles into the quadratic formula, we solved for \( b \) and \( c \), finding them to be 1 and -1, respectively. This gives us a complete equation \( y = x^2 + x - 1 \) representing this specific sequence.

• General form: \( y = ax^2 + bx + c \)
• Parabolically represented relationships
• Derived through consistent second differences

Patterns and Sequences in Mathematics

Mathematical patterns and sequences allow us to predict and understand recurring structures in numbers. They are composed of terms that relate either through operations or positions. The sequence of numbers from our exercise fits a particular pattern, identifiable through the order and combinations of their differences. Recognizing that a sequence behaves quadratically means that there is a particular rule or formula behind the generation of numbers. Discovering such rules is crucial. It helps us solve for unknown terms and predict future numbers in that sequence. In this case, our sequence of tiles follows a quadratic pattern which we verified through the second differences. Once verified, using the quadratic equation \( y = x^2 + x - 1 \), we predicted the 12th term in the sequence. Understanding sequences and predicting outcomes using these patterns is a foundational skill in math.

• Recognize patterns through operations or positions
• Quadratic sequences based on constant second differences
• Predict future terms with developed formulas