Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. h(x) = (1/2) (x − 1)² – 1

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Start with the standard quadratic function f(x) = x². This is a parabola that opens upwards with its vertex at the origin (0, 0).

Identify the transformations applied to f(x) = x² to obtain h(x) = (1/2)(x − 1)² − 1. The transformations include: (1) horizontal shift, (2) vertical stretch/compression, and (3) vertical shift.

First, note the horizontal shift. The term (x − 1)² indicates a shift of the graph 1 unit to the right. This is because the subtraction inside the parentheses moves the graph in the opposite direction of the sign.

Next, observe the vertical stretch/compression. The coefficient (1/2) in front of (x − 1)² compresses the graph vertically by a factor of 1/2. This makes the parabola wider compared to the standard f(x) = x².

Finally, apply the vertical shift. The term −1 at the end of the function shifts the entire graph downward by 1 unit. Combine all these transformations to graph h(x). Start by shifting the vertex to (1, -1), then apply the vertical compression and plot the new parabola.

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주요 개념

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Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + 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 the standard quadratic function, f(x) = x², is essential for applying transformations to graph other quadratic functions.

Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a function. For quadratic functions, common transformations include vertical and horizontal shifts, which are determined by adding or subtracting values from the input (x) or output (f(x)). In the given function h(x) = (1/2)(x − 1)² – 1, the graph is shifted right by 1 unit and down by 1 unit, while also being vertically compressed by a factor of 1/2.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is expressed as f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form makes it easier to identify the vertex and understand the transformations applied to the standard quadratic function. In the function h(x) = (1/2)(x − 1)² – 1, the vertex is at the point (1, -1), indicating the lowest point of the parabola due to the positive leading coefficient.