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

Verified step by step guidance

Step 1: Understand the equation y = x^2 + 2. This is a quadratic equation, which represents a parabola that opens upwards because the coefficient of x^2 is positive.

Step 2: Create a table of values for x and y. Use the given x-values (-3, -2, -1, 0, 1, 2, 3). For each x-value, substitute it into the equation y = x^2 + 2 to calculate the corresponding y-value.

Step 3: For example, when x = -3, substitute into the equation: y = (-3)^2 + 2. Similarly, calculate y for all other x-values (-2, -1, 0, 1, 2, 3).

Step 4: Plot the points (x, y) on a coordinate plane. Each point corresponds to an x-value and its calculated y-value. For example, if x = -3 and y = 11, plot the point (-3, 11). Repeat for all other points.

Step 5: Connect the plotted points with a smooth curve to form the graph of the parabola. Ensure the curve is symmetric about the y-axis, as the equation y = x^2 + 2 is symmetric.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Graphing Points

Graphing points involves plotting specific coordinates on a Cartesian plane. For the equation y = x^2 + 2, you will substitute the given x-values (-3, -2, -1, 0, 1, 2, 3) into the equation to find the corresponding y-values. This process helps visualize the relationship between x and y, forming the curve of the quadratic function.

Vertex and Axis of Symmetry

The vertex of a parabola is the highest or lowest point on the graph, depending on its orientation. The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves, passing through the vertex. For the function y = x^2 + 2, the vertex can be found at the point (0, 2), which is crucial for accurately sketching the graph.

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