Curve Analysis: Intersections, Symmetry, and Asymptotes
Intersection of a Curve with Coordinate Axes
To find where a curve intersects the coordinate axes (i.e., the points where it crosses the axes), substitute one of the variables representing an axis with zero. Then, solve the resulting equation to find the values where the curve intersects that axis. Repeat this process for the other axis.
| Example | |
Determine the points where the curve intersects the x and y axes. Identify the type of curve it represents.
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| Steps | Procedure |
Identify the type of curve: | By observing the equation, we can see that both coefficients of the quadratic terms are positive and have the same value, indicating the equation represents a circle. |
Substitute x = 0 into the equation to find the points where the curve intersects the y axis. |
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Apply the quadratic formula to solve for y: |
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Substitute y = 0 into the equation to find the points where the curve intersects the x axis. |
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Apply the quadratic formula to solve for x: |
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| Answer | |
The curve is a circle, and the intersections with the axes are:
and
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Symmetry of a Curve
A curve is symmetric with respect to a symmetry axis when for each point on the curve, there is a corresponding point such that both points are symmetric to the axis.
Symmetry Regarding the x Axis
Theorem If the equation of a curve remains unchanged when the variable y is replaced with -y, the curve is symmetric with respect to the x axis. |
Symmetry Regarding the y Axis
Theorem If the equation of a curve remains unchanged when the variable x is replaced with -x, the curve is symmetric with respect to the y axis. |
A curve is symmetric with respect to a center of symmetry O when for each point on the curve, there is another corresponding point such that both points are symmetric with respect to O.
Symmetry Regarding the Origin
Theorem If the equation of a curve remains unchanged when the variables x and y are replaced with -x and -y respectively, the curve is symmetric with respect to the origin. |
| Example | |
Determine if the following curve is symmetric with respect to any of the coordinate axes. x2-3y=0 | |
| Steps | Procedure |
First, substitute x with -x to see if the equation is altered. |
Since the original equation is not altered, the curve is symmetric with respect to the y axis. |
Now, substitute y with -y in the equation. |
Since the original equation is altered in the sign of the second term, the curve is not symmetric with respect to the x axis. |
| Answer | |
The curve is symmetric with respect to its y axis and is graphed as:
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Asymptotes of a Curve
If, for a certain curve, there is a line such that as a point on the curve moves indefinitely far away from the origin, the distance from that point to the line continuously decreases and approaches zero, this line is known as an asymptote of the curve.
When a curve has an asymptote, it is open. There are three types of asymptotes:
- Horizontal asymptotes: lines parallel to or coinciding with the x axis.
- Vertical asymptotes: lines parallel to or coinciding with the y axis.
- Oblique asymptotes: lines that are not parallel to any of the coordinate axes.
To find the asymptotes of a curve, solve for y in terms of x and solve for x in terms of y. Analyze for which values of x or y the other variable is undefined.
| Example | |
Determine the asymptotes for the curve:
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| Steps | Procedure |
From observing the equation, it is determined that the function is undefined for values of x that make the denominator equal to zero. |
Therefore, the curve has two vertical asymptotes: x = 1 and x = -1. |
To verify if the curve has horizontal asymptotes, solve for x as a function of y: |
This equation is defined only if the denominator is not zero, so the horizontal asymptote is y = 2. |
| Answer | |
The asymptotes of the curve are: Vertical: x = 1 and x = -1 Horizontal: y = 2 |
