91Ó°ÊÓ

Quadratic functions are a fundamental part of algebra and are characterized by their highest degree term being squared. They generally take the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In the exercise presented, the function is \( f(x) = x^2 + 2x \). This specific function has a quadratic term, \( x^2 \), and a linear term, \( 2x \), with \( a = 1 \), \( b = 2 \), and \( c = 0 \).

Quadratic functions often have parabolic graphs which open upwards if \( a > 0 \) and downwards if \( a < 0 \). The vertex of the parabola, the turning point, plays a crucial role in the behavior of the function. Additionally, the line of symmetry can be found at \( x = -\frac{b}{2a} \), which in this particular case is \( x = -1 \). Understanding these properties helps in graphing the function and analyzing its behavior over different intervals.

Function values are outputs that we get after substituting specific inputs into a function. In the case of evaluating a function, this means plugging an \( x \)-value into the function \( f(x) \). For the function \( f(x) = x^2 + 2x \), when we substitute \( x = 3 \), we get \( f(3) = 3^2 + 2 \times 3 = 15 \). Similarly, when substituting \( x = 5 \), we get \( f(5) = 5^2 + 2 \times 5 = 35 \).

These calculations are vital as they allow us to determine specific characteristics of the function, such as its average rate of change over an interval. It's quite important to correctly compute these values, as they serve as inputs into further calculations like determining rates of change or locating intersections.

Interval notation provides a concise method to describe intervals on the number line. It's an essential concept to specify domains and ranges of functions, as well as intervals of interest for calculations. For example, the interval \([3, 5]\) indicates all numbers \( x \) where \( 3 \leq x \leq 5 \). The square brackets \([\ ]\) mean that both endpoints are included in the interval, also known as a closed interval.

Understanding interval notation is crucial for defining where a function behaves in certain ways, such as increasing or decreasing, and it's commonly used in calculus for defining intervals of differentiation or integration. When computing average rates of change, the interval specifies the \( x \)-values over which the calculation is to be performed, impacting the function values used in these operations. Utilizing interval notation accurately ensures clear communication of these mathematical concepts.