Problem 1 - Algebra Form and Function

Question

Find the average rate of change of \(f(x)=x^{2}+3 x\) on the intervals indicated. Between 2 and 4 .

Step-by-Step Solution

Answer

Answer: The average rate of change of the function \(f(x) = x^2 + 3x\) on the interval [2, 4] is 9.

Step 1: Calculate f(a) and f(b)

First, let's find the function values at the endpoints of our interval: \(f(a) = f(2) = (2)^2+3(2) = 4+6=10\) \(f(b) = f(4) = (4)^2+3(4) = 16+12=28\)

Step 2: Use the formula to find the average rate of change

Now, let's plug these function values and the endpoints into our formula to find the average rate of change: Average rate of change = \(\frac{f(4) - f(2)}{4-2} = \frac{28 - 10}{2} = \frac{18}{2} = 9\) The average rate of change of \(f(x) = x^2+3x\) on the interval [2, 4] is 9.

Key Concepts

Quadratic Functions

Quadratic functions are mathematical expressions of the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. In these functions, the highest power of \( x \) is 2, making them parabolas when graphed. Quadratics can open upwards or downwards depending on the sign of \( a \):

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), it opens downwards.

The quadratic function discussed here is \( f(x) = x^2 + 3x \). Notice how there is no constant term \( c \), making it slightly simpler. However, it still possesses all characteristics of a quadratic. This function will graph as a parabola.
Key features of quadratic functions include their vertex, axis of symmetry, and whether they have minima or maxima. In our specific case, we are interested in evaluating function values at specific points on the parabola.

Intervals in Algebra

In algebra, intervals define specific sections over which we evaluate functions or analyze certain characteristics such as growth, behavior, or rate of change. An interval is often denoted by two endpoints and can be closed \([a, b]\), open \((a, b)\), or half-open \([a, b)\) or \((a, b]\). Each notation provides information about which endpoints are included in the interval.
In the context of this problem, we're evaluating the function \( f(x) = x^2 + 3x \) between \( x=2 \) and \( x=4 \), forming a closed interval \([2,4]\). This means both 2 and 4 are included in our analysis.

• A closed interval like \([2,4]\) will account for values at both \( x=2 \) and \( x=4 \).
• Such intervals are often used to find things like average rates of change, as borders clearly define the region of interest.

Function Evaluation

Function evaluation involves determining the value of a function for specific inputs. In mathematical notation, if you have \( f(x) \), and wish to evaluate it at \( x = a \), you simply substitute \( a \) in place of \( x \) in the function expression.
For the quadratic function in our exercise, we computed \( f(2) \) and \( f(4) \) by inserting these values into \( f(x) = x^2 + 3x \):

• At \( x=2 \), \( f(2) = (2)^2 + 3(2) = 10 \).
• At \( x=4 \), \( f(4) = (4)^2 + 3(4) = 28 \).

Each calculation provides a specific point on the parabola. Function evaluation is a fundamental skill in algebra as it allows you to explore the behavior of functions over different domains.