Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=x²+2

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Identify the type of function given. Since the function is \(f(x) = x^2 + 2\), it is a quadratic function, which graphs as a parabola.

Recognize the basic shape of the graph. The parent function is \(y = x^2\), which is a parabola opening upwards with its vertex at the origin \((0,0)\).

Determine the effect of the '+ 2' in the function. This constant shifts the entire graph of \(y = x^2\) vertically upward by 2 units.

Find the vertex of the parabola. Since the parent function's vertex is at \((0,0)\) and the graph is shifted up by 2, the new vertex is at \((0, 2)\).

Plot several points by choosing values for \(x\), calculating \(f(x) = x^2 + 2\), and then sketch the parabola opening upwards with vertex at \((0, 2)\).

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Quadratic Functions

A quadratic function is a polynomial function of degree two, typically written as f(x) = ax² + bx + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. Understanding the shape and properties of quadratic functions is essential for graphing them accurately.

Vertex of a Parabola

The vertex is the highest or lowest point on the graph of a quadratic function, representing its maximum or minimum value. For f(x) = x² + 2, the vertex is at (0, 2), since the function is in vertex form with no linear term. Identifying the vertex helps in sketching the parabola correctly.

Graphing Techniques for Quadratic Functions

Graphing a quadratic involves plotting the vertex, finding additional points by substituting x-values, and using symmetry about the axis of symmetry (x = -b/2a). Recognizing shifts, such as the +2 in f(x) = x² + 2, indicates a vertical translation of the parabola upward by 2 units.

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=x2+2