Lecture 3
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The document discusses translating coordinate axes by shifting the old axes horizontally by h units and vertically by k units to obtain new parallel axes. When translating axes, the new coordinates (x', y') of any point P can be determined by adding h to the old x-coordinate and adding k to the old y-coordinate. Translating axes allows equations to be simplified by changing the reference coordinate system.
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Lecture 15
1. The curve passes through the origin as there is no constant term in the equation. To find the tangents at the origin, the lowest degree terms are equated to zero, giving the x-axis as a tangent.
2. The curve meets the coordinate axes only at the origin. There are no other points of intersection.
3. The curve is symmetric about the x-axis as the powers of y in the equation are even. There are no other symmetries.
4. The y-axis is the only asymptote, obtained by equating the coefficient of the highest power of y to zero. There is no asymptote parallel to the x-axis.
5. The curve exists in
Lecture 10
Parametric equations define a curve where x and y are defined in terms of a third variable, called the parameter. The document provides examples of parametric equations and how to eliminate the parameter to obtain a single equation relating x and y, whose graph is the same as that defined by the original parametric equations. Examples shown include linear, circular and parabolic curves. Exercises are provided to eliminate parameters from additional parametric equations and sketch the resulting curves.
Lecture 11
This document discusses rational functions and how to find their asymptotes. It defines a rational function as a function of the form f(x)=p(x)/q(x) where p and q are polynomials. It then explains that vertical asymptotes occur where the denominator equals 0, and how to determine if there is a horizontal or oblique asymptote based on comparing the degrees of the numerator and denominator polynomials. Specifically, if the degree of the numerator is less than the denominator, the horizontal asymptote is the x-axis; if they are equal, the horizontal asymptote is y=leading coefficient of the numerator/leading coefficient of the denominator; and if the numerator degree is greater, there is an oblique asymptote
Lecture 12
This document defines and discusses absolute and local extreme values of functions. It states that a continuous function on a closed interval will have both an absolute maximum and minimum value. Absolute extrema can occur at endpoints or interior points, while local extrema occur at interior points where the derivative is 0 or undefined. To find the absolute extrema on an interval, evaluate the function at the endpoints and critical points, and the greatest and least values will be the absolute max and min.
Lecture 12
This document defines and discusses absolute and local extreme values of functions. It states that a continuous function on a closed interval will have both an absolute maximum and minimum value. Absolute extrema can occur at endpoints or interior points, while local extrema occur at interior points where the derivative is 0 or undefined. To find the absolute extrema on an interval, evaluate the function at the endpoints and critical points, and the greatest and least values will be the absolute max and min.
Lecture 11
This document discusses rational functions and how to find their asymptotes. It defines a rational function as a function of the form f(x)=p(x)/q(x) where p and q are polynomials. It explains that vertical asymptotes occur where the denominator equals 0, and horizontal or oblique asymptotes depend on whether the degree of the numerator is less than, equal to, or greater than the degree of the denominator. The document provides examples and step-by-step instructions for determining all types of asymptotes of rational functions.
Lecture 10
Parametric equations define a curve where x and y are defined in terms of a third variable, called the parameter. The document provides examples of parametric equations and how to eliminate the parameter to obtain a single equation relating x and y, whose graph is the same as that defined by the original parametric equations. Examples shown include linear, circular and parabolic curves. Exercises are provided to eliminate parameters from additional parametric equations and sketch the resulting curves.
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- 1. Translation of Axes Prepared by: Prof. Teresita P. Liwanag – Zapanta B.S.C.E., M.S.C.M., M.Ed. (Math-units), PhD-TM (on-going)
- 2. SPECIFIC OBJECTIVES: At the end of the lesson, the student is expected to be able to: • translate coordinate axes. • find new coordinates after translation. • find new equation if the origin is translated to a given point.. • simplify the equations by translation of axes. • graph the transformed equation.
- 3. Definition: Translation of Axes Consider a transformation in which the new axes are parallel to the original axes and similarly directed. Translation of axes is related to performing two geometric transformations: a horizontal shift and a vertical shift. Hence the new axes can be obtained by shifting the old axes h units horizontally and k units vertically while keeping their directions unchanged. Let x and y stand for the coordinates of any point P when referred to the old axes, and x’ and y’ the coordinates of P with respect to the new axes. Then x = x’ + h and y = y’ + k.