Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3
y = 9 - x²

Verified step by step guidance

1. Step 1: Understand the equation. The given equation is y = 9 - x^2. This is a quadratic equation, and its graph will be a parabola. The term -x^2 indicates that the parabola opens downward because the coefficient of x^2 is negative.
2. Step 2: Create a table of values. Substitute the given x-values (-3, -2, -1, 0, 1, 2, 3) into the equation y = 9 - x^2 to calculate the corresponding y-values. For example, when x = -3, y = 9 - (-3)^2 = 9 - 9 = 0. Repeat this process for all x-values.
3. Step 3: Plot the points. Once you have the table of values, plot the points (x, y) on a coordinate plane. For example, if x = -3 gives y = 0, plot the point (-3, 0). Repeat for all other x-values.
4. Step 4: Draw the graph. After plotting all the points, connect them smoothly to form the parabola. Ensure the curve is symmetric about the y-axis, as the equation y = 9 - x^2 is symmetric with respect to the y-axis.
5. Step 5: Analyze the graph. The vertex of the parabola is at (0, 9), which is the maximum point since the parabola opens downward. The x-intercepts are the points where y = 0, and the y-intercept is the point where x = 0. Verify these features on your graph.

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 'a'. Understanding the shape and properties of parabolas is essential for graphing equations like y = 9 - x^2.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on its graph, depending on its orientation. For the equation y = 9 - x^2, the vertex is 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 are then plotted to visualize the parabola. This process is crucial for accurately representing the function's graph.