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To understand the behavior of a function, one of the most effective tools is its graph. The graph of a function gives a visual representation of all the possible points \( (x, f(x)) \) that satisfy the function's equation. For a continuous function, you can draw the graph without lifting the pen from the paper. When graphing \( f(x) = 9 - x^2 \), a parabolic shape is revealed. The vertex of this parabola is at (0, 9), and it opens downwards because the coefficient of \( x^2 \) is negative.

By graphing, we obtain an overview of how the function behaves across different values of \( x \), identifying maximum points, intercepts, and intervals of increase or decrease. This visual insight often makes it easier to determine the limit of the function at a particular point, as it shows how the y-values change as \( x \) approaches a specific value from both the left and the right.

Parabolic functions, which are also known as quadratic functions, are of the form \( f(x) = ax^2 + bx + c \). The graph of a parabolic function is a parabola. Depending on the sign of \( a \), the parabola opens upwards (when \( a \) is positive) or downwards (when \( a \) is negative).

• The vertex of the parabola is the highest or lowest point on the graph and can be calculated using the formula \( x = -\frac{b}{2a} \) and then finding the corresponding \( y \) value by evaluating \( f(x) \) at that \( x \) value.
• The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves.
• The y-intercept occurs when you set \( x \) to 0, which gives the point (0, \( c \) ).
• The x-intercepts (if they exist) are the points at which the parabola crosses the \( x \) -axis, found by setting the function equal to zero and solving for \( x \) .

Understanding these characteristics helps in sketching the graph and anticipating the behavior of the function, such as the limit we are trying to find.

In mathematics, the limit describes the value that a function approaches as the input (or 'independent variable') approaches some value. Limits are central to calculus and are utilized to define continuity, derivatives, and integrals.

When evaluating the limit of a function at a point \( x = a \) , we are looking at the behavior of \( f(x) \) as \( x \) gets arbitrarily close to \( a \) from both the left ( \( f(x) \) when \( x < a \) ) and the right ( \( f(x) \) when \( x > a \) ). If the values of \( f(x) \) approach the same number from both sides, then the limit exists and is equal to that number. However, if the values do not approach a single number, or if the function does not approach a value at all, then the limit does not exist. In the case of \( f(x) = 9 - x^2 \) at \( x=-2 \), we find through evaluation that the limit exists and equals 5, as \( f(x) \) approaches the value of 5 from both sides of -2.

Evaluating a function means finding the output value \( f(x) \) for a given input \( x \) . To evaluate \( f(x) = 9 - x^2 \) at \( x = -2 \), substitute \( -2 \) for every instance of \( x \) in the function equation and simplify: \( f(-2) = 9 - (-2)^2 = 9 - 4 = 5 \). This tells us the exact value of the function at that point.

Often, simply knowing the function value at one point isn't enough, particularly if there's a need to understand the overall behavior—this is where graphing and limits come into play. Nonetheless, the skill of evaluating a function is crucial, not only for computing specific values but also as part of the process of graphing functions and determining limits analytically. The evaluation step is a key component in the overall understanding of how a function behaves across its entire domain.