Describe how the graph of y^2-(x^2)/(k^2)=1 changes as |k| increases.

Name: Problem 39, Chapter 10: Algebra 2
Uploaded: 2020-04-13T15:08:22-08:00
Duration: 2 min 5 s
Channel: Charles Carter
Description: Describe how the graph of y^2-(x^2)/(k^2)=1 changes as |k| increases.

step 1 This problem wants us to just talk about how if we have the hyperbola y squared minus x squared

Describe how the graph of y^2-(x^2)/(k^2)=1 changes as |k|...

Key Concepts

Asymptotes

Asymptotes are straight lines that the branches of a hyperbola approach but never actually intersect. For a hyperbola in the form y²/1 - x²/(k²) = 1, the equations of the asymptotes are given by y = ±(1/|k|)x. These lines help determine the graph's direction for large values of x and y. Understanding asymptotes is crucial because they reveal the trend of the hyperbola's curvature and how the graph 'opens' as parameters vary.

Parameter Variation Effects

The parameter k in the equation influences the horizontal scaling of the hyperbola. As the absolute value of k increases, the term k² in the denominator grows, resulting in a decrease in the slopes of the asymptotes (since the slope is 1/|k|). This means that the branches of the hyperbola become less steep and spread out more horizontally, altering the overall opening of the graph while keeping the vertical distance between vertices constant. Recognizing this effect is key to predicting and understanding the transformations in the hyperbola's shape as k varies.

Standard Form Representation

Writing the equation in the standard form clarifies the role of each parameter. In the form y²/1 - x²/(k²) = 1, the value '1' corresponds to the square of the distance from the center to the vertices along the y-axis (since the hyperbola opens vertically), while k² is associated with the horizontal spread of the graph. This representation is essential for identifying the key features of the hyperbola and understanding how variations in parameters affect its geometry.

Hyperbola

A hyperbola is one of the conic sections, defined as the set of points where the absolute difference of the distances from two fixed foci is constant. Its graph consists of two separate branches that open in opposite directions. When expressed in standard form, the hyperbola clearly displays its symmetry, vertices, and orientation, helping in understanding its overall shape and behavior as parameters change.

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