Question

One measurement of the quality of a quarterback in the National Football League is known as the quarterback rating. The rating formula is $$R(c, t, i, y)=\frac{50+20 c+80 t-100 i+100 y}{24}.$$ percentage of passes completed, \(t\) is the percentage of passes thrown for touchdowns, \(i\) is the percentage of intercepted passes, and \(y\) is the yards gained per attempted pass.a. In his career, Hall of Fame quarterback Johnny Unitas completed \(54.57 \%\) of his passes, \(5.59 \%\) of his passes were thrown for touchdowns, \(4.88 \%\) of his passes were intercepted, and he gained an average of 7.76 yards per attempted pass. What was his quarterback rating? b. If \(c, t,\) and \(y\) remained fixed, what happens to the quarterback rating as \(i\) increases? Explain your answer with and without mathematics.

Short answer

Answer: When the number of intercepted passes increases, the quarterback rating will decrease. This is because the quarterback rating formula is inversely proportional to the number of intercepted passes and more intercepted passes indicate more mistakes from the quarterback, lowering their overall effectiveness and performance.

Step-by-step solution

Understand the formula

The quarterback rating formula is given by $$R(c, t, i, y)=\frac{50+20 c+80 t-100 i+100 y}{24}.$$ where \(c\) is the percentage of passes completed, \(t\) is the percentage of passes thrown for touchdowns, \(i\) is the percentage of intercepted passes, and \(y\) is the yardage gained per attempted pass. We will use this formula to calculate Johnny Unitas's quarterback rating.

Substitute the given values into the formula

We know that for Johnny Unitas, \(c=54.57 \%\), \(t=5.59 \%\), \(i=4.88 \%\), and \(y=7.76\). Substituting these into the formula, we get: $$R(54.57, 5.59, 4.88, 7.76) = \frac{50 + 20(54.57) + 80(5.59) - 100(4.88) + 100(7.76)}{24}.$$

Calculate the quarterback rating

Now, we will calculate the values within the formula: $$R(54.57, 5.59, 4.88, 7.76) = \frac{50 + 1091.4 + 447.2 - 488 + 776}{24} = \frac{1818.4}{24} = 75.767.$$ So, Johnny Unitas's quarterback rating is 75.767.

Analyzing the effect of intercepted passes on the quarterback rating

We want to find out what happens to the quarterback rating when \(i\) increases, while \(c\), \(t\), and \(y\) remain constant. Looking at the quarterback rating formula, we can observe that the term involving \(i\) is negative (-100 i) which means that as \(i\) increases, the value of the entire expression decreases. Mathematically, since the formula is a linear function of \(i\), this means that the quarterback rating is inversely proportional to the number of intercepted passes \(i\). As the number of intercepted passes increases, the quarterback rating will decrease. Now, we can explain this without mathematics: As a quarterback throws more intercepted passes, they are making more mistakes, resulting in a decrease in their overall effectiveness and performance. This is reflected in their quarterback rating, which will become lower as the number of intercepted passes increases.

Key concepts

Linear Function

In the context of calculating a quarterback's rating, understanding the concept of a linear function is crucial. A linear function is a mathematical expression that creates a straight line when graphed. This function has a constant rate of change, often referred to as the slope.

The quarterback rating formula is a prime example of a linear function. It is structured as follows:

• The formula is \[ R(c, t, i, y)=\frac{50+20c+80t-100i+100y}{24} \]
• Each term: \(20c\), \(80t\), \(-100i\), and \(100y\), involves multiplying a variable by a constant coefficient, which represents the rate of change that each component contributes to the rating.
• The constant "+50" adjusts the starting point of the rating.

This linear structure means any change in the values of \(c\), \(t\), \(i\), or \(y\) leads to a proportional adjustment in the rating. This makes it an efficient way to quickly derive insights about a quarterback's performance based on specific game statistics.

Inverse Proportionality

Inverse proportionality plays a significant role in understanding how changes in one particular statistic, like interceptions, affect the quarterback rating. In mathematics, when two variables are inversely proportional, one goes up as the other goes down.

In the formula:

• The term \(-100i\) indicates this inverse relationship between interceptions \(i\) and the overall rating \(R\).
• As \(i\) increases, the value of \(-100i\) becomes more negative, decreasing the total in the numerator of the formula. This, in turn, lowers the resulting rating \(R\).

Given this setup, quarterbacks aiming for a higher rating should strive to minimize their interception rate. This shows that the fewer mistakes made in terms of intercepted passes, the higher their effectiveness, as captured by the quarterback rating. Understanding this inverse proportionality helps athletes focus on areas that can significantly influence their performance metrics.

Sports Statistics

In sports, particularly in American football, statistics are not just numbers—they tell stories and measure performance. The quarterback rating is an example of this, combining multiple performance metrics into a single value that helps assess a player's overall ability.

Components of the rating include:

• Completion percentage \(c\), reflecting the accuracy and efficiency of a quarterback's throws.
• Touchdowns \(t\), representing scoring ability and effectiveness in critical, game-changing situations.
• Interceptions \(i\), indicating errors or misjudgments leading to turnovers and lost opportunities.
• Yards per attempt \(y\), showing the ability to move the ball downfield effectively.

By analyzing these statistics collectively, coaches, analysts, and fans can gain a comprehensive view of a quarterback's strengths and weaknesses. This helps in strategizing and making informed decisions to improve team performance.