In Exercises 60–63, begin by graphing the standard quadratic function, f(x) = x^2. Then use transformations of this graph to graph the given function. g(x) = x^2 + 2

Verified step by step guidance

1. Start by graphing the standard quadratic function $f(x)=x^2$. This is a parabola that opens upwards with its vertex at the origin (0,0).
2. Identify the transformation needed to graph $g(x)=x^2+2$. This function is a vertical shift of the standard quadratic function.
3. Recognize that the $+2$ in $g(x)=x^2+2$ indicates a vertical shift upwards by 2 units.
4. Shift the entire graph of $f(x)=x^2$ upwards by 2 units. This means every point on the graph of $f(x)$ is moved 2 units higher.
5. The new graph, $g(x)=x^2+2$, is a parabola that opens upwards with its vertex now at (0,2).

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which opens upwards if 'a' is positive and downwards if 'a' is negative. Understanding the basic shape and properties of quadratic functions is essential for analyzing their transformations.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For quadratic functions, vertical shifts occur when a constant is added or subtracted from the function, such as in g(x) = x^2 + 2, which shifts the graph of f(x) = x^2 upward by 2 units. Recognizing these transformations helps in accurately graphing modified functions.

Standard Form of a Quadratic Function

The standard form of a quadratic function is f(x) = ax^2 + bx + c, where 'a', 'b', and 'c' are constants. This form allows for easy identification of the vertex and the direction of the parabola. In the context of transformations, understanding how changes in 'c' affect the graph's position is crucial for accurately graphing functions like g(x) = x^2 + 2.