91Ó°ÊÓ

Quadratic functions are a type of polynomial function with the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. In our exercise, the function is \( g(t) = -4.9 t^2 \). This is a simple quadratic function with no linear term (\( bx \)) or constant term (\( c \)).

Quadratic functions are characterized by their parabolic shape when graphed. If the coefficient of \( x^2 \) (in this case, \( -4.9 \)) is negative, the parabola opens downwards, meaning it curves downwards like an upside-down U.

Key properties of quadratic functions include the vertex, which is the highest or lowest point of the parabola, and the axis of symmetry, which is a vertical line passing through the vertex. These functions are particularly important in physics for modeling projectile motion and other phenomena involving acceleration.

The substitution method is a fundamental technique used to evaluate functions at specific points. It involves replacing the variable in the function with a given value. Let's break down the steps used in our exercise:

• To evaluate \( g(1) \), we substitute \( t = 1 \) into \( g(t) = -4.9 t^2 \). This gives us \( g(1) = -4.9 \times 1^2 = -4.9 \).
• To evaluate \( g(\pi) \), we substitute \( t = \pi \) into the function, resulting in \( g(\pi) = -4.9 \times \pi^2 \).
• To evaluate \( g\left(\frac{1}{\sqrt{4.9}}\right) \), we substitute \( t = \frac{1}{\sqrt{4.9}} \) to get \( g\left(\frac{1}{\sqrt{4.9}}\right) = -4.9 \left(\frac{1}{\sqrt{4.9}}\right)^2 \) which simplifies to \( -1 \).

Substitution is useful because it simplifies the function evaluation process, allowing us to find specific values for given inputs rapidly. This method is widely applied in calculus, algebra, and even in solving systems of equations in other contexts.

Mathematical constants are numbers with fixed values that arise naturally in various mathematical contexts. Some well-known constants include \( \pi \) and \( e \).

In this exercise, we encounter the constant \( \pi \), approximately equal to 3.14159. \( \pi \) represents the ratio of the circumference of a circle to its diameter and is essential in trigonometry, geometry, and calculus.

Another constant involved here is the specific number \( 4.9 \). While not universally recognized like \( \pi \), it is significant in certain physics equations, where it often appears in the context of gravitational acceleration (particularly in metric units, \( g/2 \), where \( g \approx 9.8 \ m/s^2 \)).

Understanding these constants and their roles makes it easier to grasp complex mathematical and physical concepts. For example, seeing \( 4.9 \) immediately in the context of physics might remind you of gravity-related calculations.