Problem 19 - Algebra and Trigonometry

Question

\(17-28\) A function is given. Determine the average rate of change of the function between the given values of the variable. $$ h(t)=t^{2}+2 t ; \quad t=-1, t=4 $$

Step-by-Step Solution

Answer

The average rate of change of the function is 5.

Step 1: Understand the Average Rate of Change Formula

The average rate of change of a function between two values \(a\) and \(b\) is given by the formula:\[\frac{f(b) - f(a)}{b - a}\]Substitute \(a = -1\) and \(b = 4\) in the function \(h(t) = t^2 + 2t\).

Step 2: Calculate h(-1)

Substitute \(t = -1\) into the function to find \(h(-1)\):\[h(-1) = (-1)^2 + 2(-1) = 1 - 2 = -1\]

Step 3: Calculate h(4)

Substitute \(t = 4\) into the function to find \(h(4)\):\[h(4) = 4^2 + 2(4) = 16 + 8 = 24\]

Step 4: Apply the Average Rate of Change Formula

Insert the calculated values \(h(-1) = -1\) and \(h(4) = 24\) into the average rate of change formula:\[\frac{h(4) - h(-1)}{4 - (-1)} = \frac{24 - (-1)}{4 + 1} = \frac{25}{5} = 5\]

Key Concepts

Quadratic Functions

Quadratic functions are an essential class of polynomial functions characterized by their highest degree term being a square, or having an exponent of two. These functions are typically written in the standard form
\( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \).
The defining feature of a quadratic function is its parabolic graph, which can open either upwards or downwards, depending on the sign of \( a \). If \( a \) is positive, the parabola opens upwards, creating a U-shape, while a negative \( a \) results in a downward-opening parabola, resembling an upside-down U.

• Quadratics exhibit symmetry about a vertical axis known as the "axis of symmetry." This line passes through the vertex, the highest or lowest point on the graph.
• The vertex can be a maximum or minimum point, depending on the direction the parabola opens.
• The function's graph crosses the y-axis at the point \((0, c)\), called the y-intercept.

Understanding these properties of quadratic functions aids in visualizing changes and behaviors such as rates of change and function evaluations at specific points.

Rate of Change Formula

The rate of change formula is a crucial mathematical tool used to determine how a quantity changes over an interval. For functions, especially, it tells us the average change in the function's output per unit change in input. For a function defined by \( f(x) \), the average rate of change from \( x = a \) to \( x = b \) is given by
\[\frac{f(b) - f(a)}{b - a} \]This formula essentially computes the slope of the line, or "secant line," connecting the points \((a, f(a))\) and \((b, f(b))\) on the graph of the function.

• If the rate is positive, the function is increasing over the interval \([a, b]\).
• If the rate is negative, the function is decreasing over that interval.
• When the rate is zero, it indicates no change over the interval.

This rate of change measurement helps in various real-world applications like physics (for velocity) and economics (for cost analysis), where understanding how one variable affects another is essential.

Function Evaluation

Function evaluation is the process of calculating the output of a function given a specific input value. This process involves substituting the input value into the function's expression and performing the necessary arithmetic operations. For instance, consider the function
\( h(t) = t^2 + 2t \) from the example. To evaluate this function at \( t = -1 \), substitute -1 for \( t \):
\[h(-1) = (-1)^2 + 2(-1) = 1 - 2 = -1\]Similarly, evaluating the same function at \( t = 4 \) involves substitution as follows:
\[h(4) = 4^2 + 2(4) = 16 + 8 = 24\]Function evaluation is crucial not only for understanding how specific inputs affect outputs but also for calculating changes over intervals.

• This evaluation process is foundational in calculus, where concepts such as derivatives and integrals extend function analysis.
• Evaluating functions also aids in solving systems of equations and modeling real-world scenarios with mathematical expressions.

By practicing function evaluation, students reinforce their understanding of manipulating mathematical expressions and interpreting the results.