Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3 y = 9 - x^2

Verified step by step guidance

1. <Step 1: Understand the equation.> The given equation is a quadratic equation in the form of \( y = 9 - x^2 \). This is a downward-opening parabola because the coefficient of \( x^2 \) is negative.
2. <Step 2: Create a table of values.> Substitute each value of \( x \) from the set \{-3, -2, -1, 0, 1, 2, 3\} into the equation to find the corresponding \( y \) values.
3. <Step 3: Calculate \( y \) for each \( x \).> For example, when \( x = -3 \), substitute into the equation to get \( y = 9 - (-3)^2 \). Repeat this for each \( x \) value.
4. <Step 4: Plot the points.> Use the \( x \) and \( y \) values from your table to plot points on a coordinate plane. Each point corresponds to a pair \((x, y)\).
5. <Step 5: Draw the parabola.> Connect the plotted points with a smooth curve to form the parabola. Ensure the curve opens downwards, reflecting the negative coefficient of \( x^2 \).

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form y = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. Understanding the shape and properties of parabolas is essential for graphing quadratic equations.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on the graph, depending on whether it opens downwards or upwards. For the equation y = 9 - x^2, the vertex can be found at the point (0, 9), which is the maximum value of y. Identifying the vertex helps in sketching the graph accurately and understanding the function's behavior.

Graphing Points

Graphing points involves plotting specific values of x and their corresponding y values on a coordinate plane. In this case, substituting x values from -3 to 3 into the equation y = 9 - x^2 allows us to find the corresponding y values, which can then be plotted to visualize the quadratic function. This process is crucial for accurately representing the function's graph.