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Magazine Chapter 1: Problem 3
Determine all intercepts of the graph of the equation. Then decide whether the graph is symmetric with respect to the \(x\) axis, the \(y\) axis, or the origin. $$ x^{2}-y^{2}=1 $$
Short Answer
Expert verified
The intercepts are \((1, 0)\) and \((-1, 0)\). The graph is symmetric with respect to the \(x\)-axis, \(y\)-axis, and the origin.
Step by step solution
01
Find x-intercepts
To find the \(x\)-intercepts, we set \(y = 0\) in the equation \(x^2 - y^2 = 1\). This gives us the equation \(x^2 = 1\). Solving for \(x\), we get \(x = \pm 1\). Therefore, the \(x\)-intercepts are \((1, 0)\) and \((-1, 0)\).
02
Find y-intercepts
To find the \(y\)-intercepts, we set \(x = 0\) in the equation \(x^2 - y^2 = 1\). This gives us the equation \(-y^2 = 1\), which has no real solutions because the square of a real number cannot be negative. Therefore, there are no \(y\)-intercepts.
03
Test for symmetry with respect to the x-axis
To test for symmetry with respect to the \(x\)-axis, we replace \(y\) with \(-y\) in the equation and check if the equation remains unchanged. Substituting, we have \(x^2 - (-y)^2 = x^2 - y^2 = 1\), so it is unchanged. Therefore, the graph is symmetric with respect to the \(x\)-axis.
04
Test for symmetry with respect to the y-axis
To test for symmetry with respect to the \(y\)-axis, we replace \(x\) with \(-x\) in the equation and check if the equation remains unchanged. Substituting, we have \((-x)^2 - y^2 = x^2 - y^2 = 1\), so it is unchanged. Therefore, the graph is symmetric with respect to the \(y\)-axis.
05
Test for symmetry with respect to the origin
To test for symmetry with respect to the origin, we replace both \(x\) with \(-x\) and \(y\) with \(-y\) in the equation. Substituting, we have \((-x)^2 - (-y)^2 = x^2 - y^2 = 1\), so it is unchanged. Therefore, the graph is also symmetric with respect to the origin.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Intercepts of Equations
Intercepts are the points where a graph crosses the axes. They are crucial for understanding the behavior of a graph. In coordinate geometry, we commonly identify two types of intercepts:
For the equation \(x^2 - y^2 = 1\):
1. **x-intercepts**: Setting \(y = 0\), results in \(x^2 = 1\). Solving gives \(x = \pm 1\). Thus, the x-intercepts are \((1, 0)\) and \((-1, 0)\).2. **y-intercepts**: Setting \(x = 0\), results in \(-y^2 = 1\), which is not possible for real numbers (since a square cannot be negative), indicating no y-intercepts.
- x-intercepts: Points where the graph meets the x-axis, i.e., where y = 0.
- y-intercepts: Points where the graph meets the y-axis, i.e., where x = 0.
For the equation \(x^2 - y^2 = 1\):
1. **x-intercepts**: Setting \(y = 0\), results in \(x^2 = 1\). Solving gives \(x = \pm 1\). Thus, the x-intercepts are \((1, 0)\) and \((-1, 0)\).2. **y-intercepts**: Setting \(x = 0\), results in \(-y^2 = 1\), which is not possible for real numbers (since a square cannot be negative), indicating no y-intercepts.
Testing Symmetry
Symmetry in graphs is all about how a graph mirrors itself across specific axes or the origin. A symmetric graph can significantly simplify analysis and predictions.
There are three types of symmetry that we check for:
- **x-axis symmetry** was confirmed since \(x^2 - (-y)^2 = x^2 - y^2 = 1\), showing symmetry.- **y-axis symmetry** holds as \((-x)^2 - y^2 = x^2 - y^2 = 1\), so symmetry is present.- **Origin symmetry** exists because, \((-x)^2 - (-y)^2 = x^2 - y^2 = 1\).Thus, this graph shows symmetry about both axes and the origin!
There are three types of symmetry that we check for:
- x-axis symmetry: A graph is symmetric to the x-axis if replacing \(y\) with \(-y\) yields the same equation.
- y-axis symmetry: A graph is symmetric to the y-axis if replacing \(x\) with \(-x\) yields the same equation.
- Origin symmetry: A graph is symmetric to the origin if replacing both \(x\) with \(-x\) and \(y\) with \(-y\) yields the same equation.
- **x-axis symmetry** was confirmed since \(x^2 - (-y)^2 = x^2 - y^2 = 1\), showing symmetry.- **y-axis symmetry** holds as \((-x)^2 - y^2 = x^2 - y^2 = 1\), so symmetry is present.- **Origin symmetry** exists because, \((-x)^2 - (-y)^2 = x^2 - y^2 = 1\).Thus, this graph shows symmetry about both axes and the origin!
Hyperbolas
The equation \(x^2 - y^2 = 1\) represents a hyperbola. Hyperbolas are fascinating structures in coordinate geometry. They have two parts, called branches, opening away from each other.
Key features of hyperbolas include:
Key features of hyperbolas include:
- Two separate curves that appear like infinite mirrors of each other.
- An asymptote that the curves approach but never intersect. Asymptotes give hyperbolas their distinct shape.
- Defined by their center, vertices (closest points on each branch), and foci (points that explain the curve's shape).
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a method of analyzing geometric shapes using algebra. With its help, we can understand and visualize equations like \(x^2 - y^2 = 1\).
Main principles include:
Main principles include:
- Coordinates: Every point on a graph is represented by coordinates (x, y).
- Equations of shapes: Facilitates the study of curves such as lines, circles, parabolas, ellipses, and hyperbolas.
- Slope and distance: Calculate distances and slopes using formulas from given points.
- Intercepts and Symmetry: Understanding how graphs intersect axes and showcase symmetry is a prime focus.
