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Function evaluation is a process that involves finding the output values of a function for specific input values. It is an essential concept in algebra, particularly when dealing with quadratic equations and other types of functions. In this exercise, the function given is a quadratic equation: \[ y = 4 - x^2 \]Here, we are asked to find the value of \( y \) for a range of integer values of \( x \), starting from \(-3\) up to \(3\). Each integer in this range is plugged into the function in place of \( x \). This substitution helps us calculate the corresponding \( y \)-values.

• For \( x = -3 \), evaluate \( y = 4 - (-3)^2 \)
• For \( x = -2 \), evaluate \( y = 4 - (-2)^2 \)
• Continue this substitution process for each integer value

The goal is to see how each integer input influences the outcome, showcasing the functional relationship between \( x \) and \( y \). By evaluating the function at different points, we gain insight into its behavior.

Integer substitution in this context refers to replacing every variable \( x \) in a function with integers from a specified range, which in this case is from \(-3\) to \(3\). This allows us to determine specific output values of the function for these specific inputs.Start by taking each integer value one at a time:- Plug in \( x = -3 \) into \( y = 4 - x^2 \) to find \( y = -5 \).- Next, substitute \( x = -2 \) to determine \( y = 0 \).Each substitution provides a new point for a possible graph.To handle substitution:

• Carefully replace the variable \( x \) with an integer value
• Follow the order of operations to solve: exponents first, then subtraction

Continue this method systematically through all integer values \(-3\) to \(3\). This methodical substitution helps establish the relationship between the independent variable \( x \) and the dependent variable \( y \).

A parabola is a U-shaped curve that can open upwards or downwards on a graph. Parabolas are the graphical representation of quadratic functions like \( y = 4 - x^2 \). The vertex of this parabola is the highest or lowest point, and the axis of symmetry divides it into two symmetrical halves.For the equation \( y = 4 - x^2 \), the parabola:

• Opens downward because the coefficient of \( x^2 \) is negative
• Has a vertex at \( (0, 4) \), where the maximum value of \( y \) occurs
• Is symmetric across the y-axis

The graph of this parabola will intersect the y-axis at \( y = 4 \). Points calculated through integer substitution allow us to plot the parabola precisely. Understanding the shape and position of parabolas helps in visualizing the effect of different values of \( x \) on the function, and how it translates to the rising and falling pattern of \( y \).

Graphing functions is a crucial skill in understanding the behavior of equations like quadratics. It enables one to visually interpret how the function behaves over a range of values. For \( y = 4 - x^2 \), each evaluated point from integer substitution is plotted to form the graph of a parabola.Here's how to go about graphing the function:

• Use the calculated \( (x, y) \) pairs from integer substitution as points on a coordinate plane
• Plot each point with precise accuracy, ensuring even spacing along the x-axis
• Draw a smooth curve through the points to capture the continuous nature of the function

The resulting graph should be an even, downward-opening parabola, symmetrical about the y-axis.Graphing not only depicts the relationship between \( x \) and \( y \) but also helps identify key features like the vertex, intercepts, and symmetry. This visual method provides valuable intuition about the function's behavior across the domain.