91Ó°ÊÓ

Understanding transformations is essential when graphing rational functions. A transformation modifies the base function's graph in distinct ways. In the case of rational functions like \( f(x)=\frac{1}{x} \) or \( f(x)=\frac{1}{x^2} \), well-known transformations include shifts, stretches, compressions, and reflections.

Horizontal and Vertical Shifts

Horizontal shifts occur when we add or subtract a constant to the input value x before it is placed into the function. For instance, adding 2 inside the function to get \( f(x)=\frac{1}{(x+2)^2} \) shifts the graph 2 units left. Similarly, subtracting a number from x would shift it to the right. In the case of vertical shifts, adding or subtracting a constant outside the function would lift or lower the graph respectively.

Stretches and Compressions

Multiplying x by a number greater than 1 stretches the graph horizontally, while values between 0 and 1 compress it. Alternatively, multiplying the entire function by a number will stretch or compress the graph vertically.

Reflections

Multiplying the function or x by a negative sign causes a reflection. If x is multiplied, the graph flips over the y-axis. If the entire function is multiplied, the graph flips over the x-axis.Through understanding and applying these transformations, we can create accurate graphs for more complex functions like the one in our exercise.

A hyperbola is a type of curve that looks like two mirrored, open arcs that can either be facing sideways or up and down. The graph of a basic hyperbola function, such as \( f(x)=\frac{1}{x} \) or \( f(x)=\frac{1}{x^2} \), displays this classic, open curve shape.In the equation \( f(x)=\frac{1}{x^2} \), '1' is the coefficient, indicating the steepness of the curve, while the squared term in the denominator ensures the graph opens upwards and downwards with the vertex at the origin.

Parts of a Hyperbola Graph

• Vertex: The point where the hyperbola turns.
• Asymptotes: Lines the hyperbola approaches but never touches. For \( f(x)=\frac{1}{x^2} \), the x- and y-axes serve as asymptotes.
• Branches: The separate pieces of the hyperbola that open in opposite directions.

The hyperbolic curves get closer to the axes but never intersect them, resulting in a graph that extends indefinitely in both directions along the asymptotes, illustrating the concept of infinity in the graph's behavior.

A horizontal shift is a type of transformation that moves the graph of a function to the left or right. This is accomplished by adding or subtracting a constant from the variable x before it is inputted into the function's formula. For example, in the function \( f(x)=\frac{1}{(x+2)^2} \), a '+2' inside the parenthesis with x represents a horizontal shift of the entire graph 2 units to the left.

Visualizing Horizontal Shifts

Graphically, every point on the original curve moves horizontally in the same direction by the same amount. If x is replaced with (x - h), where h is a positive number, the graph shifts h units to the right. Conversely, if x is replaced with (x + h), the graph shifts h units to the left.

Impact on Graph's Properties

The horizontal shift does not change the shape or orientation of a graph. It only changes the location. In the context of our textbook exercise, the initial function \( f(x)=\frac{1}{x^2} \) with its vertex at the origin is shifted to the left, resulting in the new vertex at (-2, 0). Thus, the hyperbola retains its shape but is simply relocated on the coordinate plane due to the horizontal shift.