Problem 77 - Algebra and Trigonometry

Question

Find the average rate of change of \(f(x)=x^{2}-1\) from \(x_{1}=1\) to \(x_{2}=2\)

Step-by-Step Solution

Answer

The average rate of change of \(f(x)=x^{2}-1\) from \(x_{1}=1\) to \(x_{2}=2\) is 3

Step 1: Evaluate the Function at \(x_{1}\)

Substitute \(x_{1}=1\) into the function \(f(x)=x^2-1\). Thus, \(f(x_{1})=f(1)=(1)^2-1=1-1=0\)

Step 2: Evaluate the Function at \(x_{2}\)

Substitute \(x_{2}=2\) into the function \(f(x)=x^2-1\). Thus, \(f(x_{2})=f(2)=(2)^2-1=4-1=3\)

Step 3: Apply the Average Rate of Change Formula

Substitute \(f(x_{2})\), \(f(x_{1})\), \(x_{2}\), and \(x_{1}\) into the formula for the average rate of change \((f(x_{2})-f(x_{1})) / (x_{2}-x_{1})\). This results in \((f(2)-f(1)) / (2-1) = (3-0) / 1 = 3\)

Key Concepts

Quadratic Functions

Quadratic functions are fundamental in algebra and take the general form of \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. These functions create a parabolic graph that can open upwards or downwards depending on the sign of \(a\). When plotted, the highest or lowest point, called the vertex, is a significant feature of the graph.

Understanding quadratic functions is essential because they model various real-world scenarios, such as projectile motion and optimal area problems. One important property of a quadratic function is its symmetry around the vertical axis that passes through the vertex, known as the axis of symmetry.

Function Evaluation

Function evaluation involves substituting a specific value for the variable \(x\) into a function to calculate the corresponding output. In the context of the given exercise, evaluating the quadratic function \(f(x) = x^2 - 1\) at \(x_1 = 1\) and \(x_2 = 2\) is a crucial step.

By plugging in these values into the function, we obtain the outputs \(f(1)\) and \(f(2)\), which are needed to calculate the average rate of change. Such evaluations form the basis for understanding how a function behaves and are used to graph functions or find function properties like rates of change and extremities.

Rate of Change Formula

Average Rate of Change

The average rate of change for a function over an interval represents the slope of the secant line that crosses the graph at two points corresponding to the interval. The formula for finding the average rate of change is \((f(x_2) - f(x_1)) / (x_2 - x_1)\), which resembles the slope formula \((y_2 - y_1) / (x_2 - x_1)\) from coordinate geometry.

In the exercise, applying this formula to the values obtained from function evaluation at \(x_1\) and \(x_2\) yields the average rate of change from \(x = 1\) to \(x = 2\). Understanding this concept is vital in learning calculus, as this average rate approaches the instantaneous rate of change (the derivative) as the interval becomes smaller.

Algebraic Concepts

Algebraic concepts form the foundation of many areas in mathematics and are used extensively in problem-solving. They include operations with algebraic expressions, solving equations and inequalities, and understanding functions and their properties. The exercise leverages various algebraic concepts, such as function evaluation, manipulation of expressions, and understanding the behavior of quadratic functions.

Through algebra, we can generalize patterns, solve for unknowns, and express mathematical relationships. The ability to evaluate functions, as seen in the exercise, and compute rate of change are just a couple of the myriad applications of algebra in understanding and modeling the world around us.