Learn how to find the inverse of a linear function. A linear function is a function whose highest exponent in the variable(s) is 1. The inverse of a function is a function which reverses the "effect" of the original function. One important property of the inverse of a function is that when the inverse of a function is made the argument (input) of a function, the result is x.
Given a function, say f(x), to find the inverse of the function, we first change f(x) to y. Next, we change all x to y and y to x. and then we solve for y. The obtained solution for y is the inverse of the original function.
#functions #inverseoffunctions
Series Playlist https://www.youtube.com/playlist?list=PL0G-Nd0V5ZMqYo1t0Vlxd9wGfNsoRgPPg

Views: 371977
Brian McLogan

http://www.freemathvideos.com In this video series I show you how to find the reference angle of a given angel. The reference angle is the acute angle between the terminal and the horizontal axis. For each quadrant there are different formulas we will explore to find the reference angle.
θ = -145 degrees

Views: 33026
Brian McLogan

Learn how to find the slope between two points. The slope of a line is the steepness of the line. The horizontal line has a zero slope while the vertical line has an undefined slope. To determine the slope of a line passing through two points, we need to know the coordinates of 2 points on the line. Then the slope of the line is given by the quotient of the difference between the y-coordinates and the difference between the x-coordinates. i.e. slope = (y2 - y1) / (x2 - x1).
#linearequations #sloperateofchange

Views: 310404
Brian McLogan

Learn how to find the removable and non-removable discontinuity of a function. A function is said to be discontinuous at a point when there is a gap in the graph of the function at that point. A discontinuity is said to be removable, when there is a factor in the numerator which can cancel out the discontinuous factor and is said to be non-removable when there is no factor in the numerator which can cancel out the discontinuous factor.
To find the discontinuities of a rational function, it is usually useful to factor the expressions in the function and we then set the denominator equal to 0 and solve for x. The value of x for which the factor appears in both the numerator and the denominator is the point of removable discontinuity while the value of x for which the factor appears in only the denominator is the point of non-removable discontinuity.
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#rationalfunctions #findasymptotes
#rationalfunctions #findasymptotes
#graphradicals #findasymptotes

Views: 30204
Brian McLogan

Learn how to graph the equation of a circle by completing the square. Completing the square will allow us to transform the equation of a circle from general form to standard form. When the equation is in standard form we can identify the center and radius of the circle to graph to then graph the circle.
#conicsections #circleconicsections

Views: 177383
Brian McLogan

Learn about Descartes' Rule of Signs. Descartes' rule of sign is used to determine the number of positive and negative real zeros of a polynomial function. Knowing the number of positive and negative real zeros enables also to also know the number of complex zeros of a complex number.
Descartes' rule of signs states that the number of positive real zeroes in a polynomial function f(x) is the same or less than by an even number as the number of changes in the sign of the coefficients of the terms of the polynomial function. The number of negative real zeroes of the f(x) is the same as the number of changes in sign of the coefficients of the terms of f(-x) or less than this by an even number.
#polynomials #rationalzerotest #descartesruleofsigns

Views: 25269
Brian McLogan

In this video series you will learn how to solve a system of two linear equations using elimination. This is also called the addition method because we are combining two equations to eliminate a variable. Which variable you decide to eliminate is up to you but in general we choose to eliminate the variable that has the same coefficients positive and negative. Sometimes we have to multiply one or both equations by a scalar to achieve this situation. We will also explore systems with infinite many solutions and no solutions.
#systemsofequations #solvesystemsofequations

Views: 609111
Brian McLogan

Learn how to write the equation of a line given two points on the line. The equation of a line is such that its highest exponent on its variable(s) is 1. (i.e. there are no exponents on its variable(s)). There are various forms which we can write the equation of a line: the point-slope form, the slope-intercept form, the standard form, etc.
The equation of a line given two points (x1, y1) and (x2, y2) through which the line passes is given by, ((y - y1)/(x - x1)) / ((y2 - y1)/(x2 - x1)).
#linearequations #writelinearequations

Views: 80921
Brian McLogan

Learn how to solve a system of equations by substitution. To solve a system of equations means to obtain a common values of the variables that makes the each of the equation in the system true. To solve a system of equations by substitution, we solve for one of the variables in one of the equations and then substitute for the variable in the other equation and then solve for the other variable. Next we plug in the obtained value of the other variable into the equation for the initial variable we solved for to obtain the value of the variable.
The value of the two variables obtained is the solution to the system of equations.
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#systemsofequations #solvesystemsofequations

Views: 487509
Brian McLogan

Learn how to graph quadratic equations by completeing the square. A quadratic equation is an equation of the form y = ax^2 + bx + c, where a, b and c are constants. The graph of a quadratic equation is in the shape of a parabola which can either face up or down (if x is squared in the equation) or face left or right (if y is squared).
To graph a quadratic equation, we need to know some essential parts of the graph including the vertex. The vertex of a parabola is the turning point of the parabola. It is the point on the parabola at which the curve changes from increasing to decreasing or vice-versa.
Given a quadratic equation in standard form, we obtain the vertex of the equation by using the process of completing the square to rewrite the equation in the vertex form and hence extract the vertex of the parabola formed by the equation. Knowing the vertex of the graph and the parent graph, we can then apply the required transformation to obtain the required graph.
#quadraticequations #graphquadratics

Views: 381472
Brian McLogan

Learn how to write the equation of a polynomial when given rational zeros. Recall that a polynomial is an expression of the form ax^n + bx^(n-1) + . . . + k, where a, b, and k are constants and the exponents are positive integers. The zeros of a polynomial are the values of x for which the value of the polynomial is zero.
To write the equation of a polynomial, we write the given zeros in factor form and expand the product of the factors. Thus, given a, b, . . . as zeros to a polynomial, we write the equation of the polynomial by expanding the factors (x - a)(x - b) . . . = 0
#polynomials #writepolynomial

Views: 92638
Brian McLogan

Learn how to find the inverse of a rational function. A rational function is a function which has an expresion in the numerator and the denominator of the function. The inverse of a function is a function which reverses the "effect" of the original function. One important property of the inverse of a function is that when the inverse of a function is made the argument (input) of a function, the result is x.
Given a function, say f(x), to find the inverse of the function, we first change f(x) to y. Next, we change all x to y and y to x. and then we solve for y. The obtained solution for y is the inverse of the original function.
#functions #inverseoffunctions

Views: 111292
Brian McLogan

Learn how to solve absolute value equations with multiple steps. Absolute value of a number is the positive value of the number. For instance, the absolute value of 2 is 2 and the absolute value of -2 is also 2. To solve an absolute value equation where there are more terms apart from the absolute value term in the same side of the equality sign as the absolute value term, we first isolate the absolute value term. i.e. make the absolute value term the subject of the formular.
After isolating the absolute value term, we now separate the equation into the positive and the negative cases and then solve accordingly.
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#absolutevalue #solveabsequations

Views: 126496
Brian McLogan

Learn how to graph piecewise functions. A piecewise function is a function which have more than one sub-functions for different sub-intervals(sub-domains) of the function's domain. To graph a piecewice function, we graph the different sub-functions for the different sub-intervals of the function's domain.
#functions #piecewisefunctions

Views: 264458
Brian McLogan

Learn about the parts of a parabola. A parabola is the shape of the graph of a quadratic equation. A regular palabola is the parabola that is facing either up or down while an irregular parabola faces left or right. A quadratic equation is an equation whose highest exponent in the variable(s) is 2.
The parts of a parabola include: the axis of symmetry (the line passing through the vertex of the parabola to which the parabola is symmetric about), the vertex (the point at which the parabola turns), the domain (the set of possible x-values of the parabola, usually all real numbers for regular parabolas), the range (the possible y-values of the parabola which is usually the region above the vertex inclusive or below the vertex inclusive for regular parabolas), the x-intercepts (the points where the parabola cuts the x-axis) and the y-intercepts (the point(s) where the parabola cuts the y axis.
#quadraticequations #graphquadratics

Views: 99386
Brian McLogan

Learn how to evaluate the limit of a function using the difference quotient formula. The difference quotient is a measure of the average rate of change of the function over an interval, h. The limit of the difference quotient gives the derivative of the function.
The difference quotient formula states that the derivative of a function f(x) is the limit as h goes to zer0 of the quotient of the diference between f(x + h) and f(x) and h.

Views: 71540
Brian McLogan

Learn how to write the equation of a line that is perpendicular to a given line. The equation of a line is such that its highest exponent on its variable(s) is 1. (i.e. there are no exponents in its variable(s)). There are various forms which we can write the equation of a line: the point-slope form, the slope-intercept form, the standard form, etc.
When given a point (x, y) through which a line passes and the slope (m) of the line, the equation of the line is given by y - y1 = m(x - x1). For perpendicular lines, the product of their slopes is -1. Thus given the equation of a line and a point through which the line perpendicular to the given line passes, then the slope of the second line is equal to the negative of the reciprocal of the slope of the given line. Having found the slope, we can then use the formula stated above together with the given point to find the required equation.
#linearequations #writelinearequations

Views: 74411
Brian McLogan

Learn about graphing absolute value equations. An absolute value equation is an equation having the absolute value sign and the value of the equation is always positive. The graph of the parent function of an absolute value equation is a v-shaped graph starting from the origin above the x-axis and rising both sides of the y-axis and is symmetrical to the y-axis.
To graph an absolute value equation, we first graph the parent function of the absolute value equation and we then apply the necessary transformation(s) to the graph of the parent function to obtain the required graph of the absolute value equation.
#absolutevalue #graphabsequations

Views: 121232
Brian McLogan

Learn how to determine the angle between two vectors. To determine the angle between two vectors you will need to know how to find the magnitude, dot product and inverse cosine. Then, the angle between two vectors is given by the inverse cosine of the ratio of the dot product of the two vectors and the product of their magnitudes.
#trigonometry#vectors
#vectors

Views: 61060
Brian McLogan

Learn how to graph quadratics in standard form. A quadratic equation is an equation whose highest exponent in the variable(s) is 2. To graph a quadratic equation, we make use of a table of values and the fact that the graph of a quadratic is a parabola which has an axis of symmetry, to plot some points to one side of the axis of symmetry and then refrect the plotted points about the axis of symmetry.
#quadraticequations #graphquadratics

Views: 71802
Brian McLogan

Learn the basics of co-terminal angles. An angle is a figure formed by two rays which have a common endpoint. The two rays are called the sides of the angle while the common endpoint is called the vertex of the angle. We measure angles starting from the positive x-axis, i.e. the initial side of an angle measure is usually the positive x-axis. Angle measured in the anti-clockwise direction is called a positive angle while a negative angle is measured in the clockwise direction.
Two or more angles are said to be co-terminal when they have the same initial and terminal sides. Coterminal angles are found by adding/subtracting 360 degrees (for degree angle measure) or 2pi (for radian angle measure) to/from the given angle.
#trigonometry #anglesintrigonometry

Views: 40226
Brian McLogan

Learn how to write the equation of a parabola given the focus and the directrix. A parabola is the shape of the graph of a quadratic equation. A parabola can open up or down (if x is squared) or open left or right (if y is squared). Recall that the focus and the vertex of a parabola are on the same line of symmetry.
When given the focus and the directrix of a parabola, recall that the vertex of a parabola is halfway between the focus and the directrix and the focus is inside the parabola. This enables us to identify the direction which the required parabola opens. We also need to identify the value of p, which is the distance between the vertex and the focus. p is negative when the parabola opens down or left and is positive when the parabola opens right or up.
Once we identify the direction and the value of p, we can use the equation of parabola given by (y - k)^2 = 4p(x - h) for parabolas that opens up or down and (x - h)^2 = 4p(y - k) for parabolas that opens left or right.
#conicsections #parabolaconicsections

Views: 83857
Brian McLogan

Learn how to write the equation of a circle. A circle is a closed shape such that all points are equidistance (equal distance) from a fixed point. The fixed point is called the center of the circle while the distance between any point of the circle and the center of the circle is called the radius of the circle.
To write the equation of a circle, we need to know the length of the radius of the circle and the coordinate point of the center of the circle. Given a circle whose center is at (h, k) and the length of the radius is r, the equation of the circle is given by (x - h)^2 + (y - k)^2 = r^2.
#geometry #circles

Views: 128677
Brian McLogan

Learn how to write the equation of an ellipse from its properties. The equation of an ellipse comprises of three major properties of the ellipse: the major radius (a), the minor radius (b) and the center (h, k). The ellipse is vertical if the major radius in vertical and the ellipse is horizontal if the major radius is horizontal.
When given the two foci of the ellipse, the center of the ellipse is halfway between the two foci. When given the vertices of an ellipse, the major radius of an ellipse is the distance between the center of the ellipse and its vertices. Using the pythagoras identity for the relationship between the focal length (distance between the center and the foci) and the radius, we can obtain the minor radius.
After obtaining the center, the major and the minor radius, they are plugged into the equation of an ellipse to obtain the desired equation.
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#conicsections #ellipseconicsections

Views: 61796
Brian McLogan

Learn how to write the equation of a parabola given the vertex and the focus. A parabola is the shape of the graph of a quadratic equation. A parabola can open up or down (if x is squared) or open left or right (if y is squared). Recall that the focus and the vertex of a parabola are on the same line of symmetry.
When given the focus and the vertex of a parabola, recall that the focus of a parabola is inside the parabola. This enables us to identify the direction which the required parabola opens. We also need to identify the value of p, which is the distance between the vertex and the focus. p is negative when the parabola opens down or left and is positive when the parabola opens right or up.
Once we identify the direction and the value of p, we can use the equation of parabola given by (y - k)^2 = 4p(x - h) for parabolas that opens up or down and (x - h)^2 = 4p(y - k) for parabolas that opens left or right
#conicsections #parabolaconicsections

Views: 128970
Brian McLogan

Learn about Descartes' Rule of Signs. Descartes' rule of sign is used to determine the number of positive and negative real zeros of a polynomial function. Knowing the number of positive and negative real zeros enables also to also know the number of complex zeros of a complex number.
Descartes' rule of signs states that the number of positive real zeroes in a polynomial function f(x) is the same or less than by an even number as the number of changes in the sign of the coefficients of the terms of the polynomial function. The number of negative real zeroes of the f(x) is the same as the number of changes in sign of the coefficients of the terms of f(-x) or less than this by an even number.
#polynomials #rationalzerotest #descartesruleofsigns

Views: 26117
Brian McLogan

Learn how to evaluate the six trigonometric functions given some constraints. When given the value of one trigonometric function, we can use a right triangle with one of its legs on the x-axis and the other leg, perpendicular to the x-axis is drawn such that the reference angle is at the origin. The x-coordinate of the given point represents the adjacent of the reference angle while the y-coordinate represent the opposite of the reference angle. The hypothenuse is the line joining from the origin to the terminal side of the 'opposite' line.
Using the Pythagoras theorem, we can obtain any of the unknown sides of the right triangle. Having the opposite, the adjacent and the hypothenuse of the reference angle of the right triangle, we can then evaluate the other trigonometric functions accordingly within the given constraint.
#trigonometry #evaluatetrigonometricfunctions

Views: 25816
Brian McLogan

Learn how to graph linear equations written in slope intercept form. When given a linear equation in slope intercept form, (i.e. in the form y = mx + c, where m is the slope and c is the y-intercept). We first plot the y-intercept and using the slope, we can determine the rise and the run of the required line and then be able to plot the next point from the y-intercept. We then draw a straight line passing through the two plotted points.
#linearequations #graphlinearequations

Views: 167245
Brian McLogan

Learn the basics of co-terminal angles. An angle is a figure formed by two rays which have a common endpoint. The two rays are called the sides of the angle while the common endpoint is called the vertex of the angle. We measure angles starting from the positive x-axis, i.e. the initial side of an angle measure is usually the positive x-axis. Angle measured in the anti-clockwise direction is called a positive angle while a negative angle is measured in the clockwise direction.
Two or more angles are said to be co-terminal when they have the same initial and terminal sides. Coterminal angles are found by adding/subtracting 360 degrees (for degree angle measure) or 2pi (for radian angle measure) to/from the given angle.
#trigonometry #anglesintrigonometry

Views: 44896
Brian McLogan

Learn how to evaluate a piecewise function. A piecewise function is a function which uses different rules for different intervals. When evaluating a piecewise function, pay attention to the constraints of each function as you can only evaluate for the equation which falls within the constraint. i.e. when evaluating a piecewise function, first identify the part of the contraints where the argument of the function you want to evaluate falls then use the equation given for the suitable constraint to evaluate the function.
#functions #piecewisefunctions

Views: 44329
Brian McLogan

Learn how to graph a horizontal parabola. A parabola is the shape of the graph of a quadratic equation. A parabola is said to be horizontal if it opens to the left or opens to the right. A horizontal parabola results from a quadratic equation in which the y part of the equation is squared.
To sketch the graph of a parabola, we first identify the vertex, the focus and the directrix. To do this, we first write the equation in the form (y - k)^2 = 4p(x - h), where (h, k) is the vertex and p is the distance between the vertex and the focus.
After expressing the equation in the form (y - k)^2 = 4p(x - h), the vertex is given by (h, k), the focus is given by (h + p, k) and the directrix is given by the line x = h - p. After obtaining the vertex, the focus and the directrix, we can then sketch the parabola.
#conicsections #parabolaconicsections

Views: 26063
Brian McLogan

Learn how to verify trigonometric identities having rational expressions. To verify trigonometric expression means to verify that the terms on the left hand side of the equality sign is equal to the terms on the right hand side. To verify rational trigonometric identities, it is usually more convenient to start with getting rid of the denominator(s) of the rational term(s). This can be done by multiplying both the numerator and the denominator by the conjugate of the denominator, if the denominator involves addition/subtraction or by the reciprocal of the denominator, if the denominator involves product or the expression can be converted to Pythagoras trigonometric identity if possible.
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#trigidentities #simplifytrigidentities

Views: 27443
Brian McLogan

Learn how to show that two functions are inverses. The composition of two functions is using one function as the argument (input) of another function. In simple terms composition of two functions is putting one function inside another function. Thecomposition of two functions that are inverse to each other yeilds x.
To show whether two functions are inverse of each other, we composite the two functions. If the result is x, then the two functions are inverse of each other and they are not inverses otherwise.
#functions #inverseoffunctions

Views: 44017
Brian McLogan

Learn how to graph linear inequalities written in slope-intercept form. Linear inequalities are graphed the same way as linear equations, the only difference being that one side of the line that satisfies the inequality is shaded. Also broken line (dashes) is used when the linear inequality is 'excluded' (when less than or greater than is used) and a solid line is used when the inequality is 'included' (when greater than or equal to OR less than or equal to is used).
To graph a linear inequality in slope intercept form, we first plot the y-intercept and using the slope, we can determine the rise and the run of the required line and then be able to plot the next point from the y-intercept. We then draw a straight line passing through the two plotted points.
After the line representing the linear equation form of the linear inequality is drawn, we select a point either side of the line to determine which side of the line is true for the given inequality and then shade the side that satisfies the inequality.
#linearinequalities #graphlinearinequalities

Views: 27276
Brian McLogan

Learn how to determine the end behavior of the graph of a polynomial function. To do this we will first need to make sure we have the polynomial in standard form with descending powers. We will then identify the leading terms so that we can identify the leading coefficient and degree of the polynomial.
The end behavior of a polynomial depends on the leading coefficient (the coefficient of the term with the greatest power) and the degree (the exponent of the term with the greatest power) of the polynomial. If the degree is even and the leading coefficient is positive, the graph of the polynomial rises left and rises right. If the degee is even and the leading coefficient is negative, the graph of the polynomial falls left and falls right. If the degee is odd and the leading coefficient is positive, the graph of the polynomial falls left and rises right. If the degee is odd and the leading coefficient is negative, the graph of the polynomial rises left and falls right.
#polynomials #endbehavior
#polynomials #endbehavior

Views: 94748
Brian McLogan

Learn how to graph vertical ellipse not centered at the origin. A vertical ellipse is an ellipse which major axis is vertical. To graph a vertical ellipse, we first identify some of the properties of the ellipse including the major radius (a) and the minor radius (b) and the center. These properties enables us to graph the ellipse.
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#conicsections #ellipseconicsections

Views: 126935
Brian McLogan

Learn how to find the value that makes a function continuos. A function is said to be continous if two conditions are met. They are: the limit of the function exist and that the value of the function at the point of continuity is defined and is equal to the limit of the function.
To find the value that makes a function continous we evaluate the right limit, the left limit and the value of the function at the point of discontinuity and then equate the values. We then solve any resulting equation to obtain the desired value that makes the function continuos.

Views: 34893
Brian McLogan

Learn how to write the equation of a line that is parallel to a given line. The equation of a line is such that its highest exponent on its variable(s) is 1. (i.e. there are no exponents in its variable(s)). There are various forms which we can write the equation of a line: the point-slope form, the slope-intercept form, the standard form, etc.
When given a point (x, y) through which a line passes and the slope (m) of the line, the equation of the line is given by y - y1 = m(x - x1). For parallel lines, the slopes are equal. Thus given the equation of a line and a point through which the line parallel to the given line passes, then the slope of the second line is equal to the slope of the given line. Having found the slope, we can then use the formula stated above together with the given point to find the required equation.
#linearequations #writelinearequations

Views: 37527
Brian McLogan

Learn how to use the tools needed to graph a Polynomial function in standard form. The tools we will use to help us graph are end behavior, finding the zeros by factoring synthetic division as well as identify the multiplicity of each zero.
The end behavior of the polynomial can be determined by looking at the degree and leading coefficient. Once we know where the graph is going we can find the x-intercepts to obtain a general shape of the graph. We will use the multiplicity the power of each factor to determine if the graph crosses or bounces at each intercept. We use all of this information to sketch the graph.
#polynomials #graphpolynomials

Views: 107246
Brian McLogan

Learn how to graph linear inequalities written in standard form. Linear inequalities are graphed the same way as linear equations, the only difference being that one side of the line that satisfies the inequality is shaded. Also broken line (dashes) is used when the linear inequality is 'excluded' (when less than or greater than is used) and a solid line is used when the inequality is 'included' (when greater than or equal to OR less than or equal to is used).
To graph a linear inequality written in standard form, we first determine the slope and the y-intercept by rewriting the linear inequality in slope intercept form, we then plot the y-intercept and using the slope, we can determine the rise and the run of the required line and then be able to plot the next point from the y-intercept. We then draw a straight line passing through the two plotted points.
Alternatively, we can determine the x-intercept and the y-intercept of the standard form linear inequality by subtituting y = 0, then solve for x and substituting x = 0, then solve for y respectively. Recall that the x-intercept is the value of x when y = 0 and the y-intercept is the value of y when x = 0. After obtaining the values of the x-intercept and the y-intercept, we plot the points on the coordinate plane and then draw a line passing through the points.
After the line representing the linear equation form of the linear inequality is drawn, we select a point either side of the line to determine which side of the line is true for the given inequality and then shade the side that satisfies the inequality.
#linearinequalities #graphlinearinequalities

Views: 54459
Brian McLogan

Learn about the parts of a parabola. A parabola is the shape of the graph of a quadratic equation. A regular palabola is the parabola that is facing either up or down while an irregular parabola faces left or right. A quadratic equation is an equation whose highest exponent in the variable(s) is 2.
The parts of a parabola include: the axis of symmetry (the line passing through the vertex of the parabola to which the parabola is symmetric about), the vertex (the point at which the parabola turns), the domain (the set of possible x-values of the parabola, usually all real numbers for regular parabolas), the range (the possible y-values of the parabola which is usually the region above the vertex inclusive or below the vertex inclusive for regular parabolas), the x-intercepts (the points where the parabola cuts the x-axis) and the y-intercepts (the point(s) where the parabola cuts the y axis.
#quadraticequations #graphquadratics

Views: 31779
Brian McLogan

Learn all about graphing logarithmic functions. A logarithmic function is a function with logarithms in them. The graph of the parent function of a logarithmic function usually takes its domain from the positive x-axis. To graph a logarithmic function, it is usually useful to first graph the parent function (without transformations). This can be done by choosing 2-3 points from the function and plotting them on the x-y coordinate axis to see the nature of the parent function's graph.
After graphing the parent function, we then apply the given transformations to obtain the required graph. When a constant is added to x in the function, the graph of the parent function shifts to the left by the same units as the constant added to x. Similarly, when a constant is subtracted from x in the function, the graph of the parent function shifts to the right by the same units as the constant subtracted from x.
When a constant is added to the function, the graph of the parent function shifts upwards by the same units as the constant added to the function. Similarly, when a constant is subtracted from the function, the graph of the parent function shifts downwards by the same units as the constant subtracted from the function.
#graphlogarithmic #logarithms

Views: 39366
Brian McLogan

Learn how to write the equation of a parabola given the vertex and the directrix. A parabola is the shape of the graph of a quadratic equation. A parabola can open up or down (if x is squared) or open left or right (if y is squared). Recall that the focus and the vertex of a parabola are on the same line of symmetry.
When given the vertex and the directrix of a parabola, recall that the vertex of a parabola is halfway between the focus and the directrix and the focus is inside the parabola. This enables us to identify the direction which the required parabola opens. We also need to identify the value of p, which is the distance between the vertex and the focus. p is negative when the parabola opens down or left and is positive when the parabola opens right or up.
Once we identify the direction and the value of p, we can use the equation of parabola given by (y - k)^2 = 4p(x - h) for parabolas that opens up or down and (x - h)^2 = 4p(y - k) for parabolas that opens left or right.
#conicsections #parabolaconicsections

Views: 73322
Brian McLogan

Learn all about graphing exponential functions. An exponential function is a function whose value increases rapidly. To graph an exponential function, it is usually useful to first graph the parent function (without transformations). This can be done by choosing 2-3 points of the equation (including the y-intercept) and ploting them on the x-y coordinate axis to see the nature of the graph of the parent function. After graphing the parent function, we can then apply the given transformations to obtain the required graph.
#graphexponential #exponentialfunctions

Views: 140867
Brian McLogan

Learn how to graph a rational function. To graph a rational function, we first find the vertical and horizontal asymptotes and the x and y-intercepts. After finding the asymptotes and the intercepts, we graph the values and then select some random points usually at each side of the asymptotes and the intercepts and graph the points, this enables us to identify the behavior of the graph and thus enable us to graph the function.
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#rationalfunctions #graphrationalfunctions
#rationalfunctions #graphrationalfunctions

Views: 192923
Brian McLogan

Learn how to determine the end behavior of the graph of a polynomial function. To do this we will first need to make sure we have the polynomial in standard form with descending powers. We will then identify the leading terms so that we can identify the leading coefficient and degree of the polynomial.
The end behavior of a polynomial depends on the leading coefficient (the coefficient of the term with the greatest power) and the degree (the exponent of the term with the greatest power) of the polynomial. If the degree is even and the leading coefficient is positive, the graph of the polynomial rises left and rises right. If the degee is even and the leading coefficient is negative, the graph of the polynomial falls left and falls right. If the degee is odd and the leading coefficient is positive, the graph of the polynomial falls left and rises right. If the degee is odd and the leading coefficient is negative, the graph of the polynomial rises left and falls right.
#polynomials #endbehavior
#polynomials #endbehavior

Views: 74093
Brian McLogan

Learn how to solve a system of equations by substitution. To solve a system of equations means to obtain a common values of the variables that makes the each of the equation in the system true. To solve a system of equations by substitution, we solve for one of the variables in one of the equations and then substitute for the variable in the other equation and then solve for the other variable. Next we plug in the obtained value of the other variable into the equation for the initial variable we solved for to obtain the value of the variable.
The value of the two variables obtained is the solution to the system of equations.
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#systemsofequations #solvesystemsofequations

Views: 87738
Brian McLogan

Learn the basics to graphing the secant and the cosecant graphs. Recall that the secant is the reciprocal of the cosine while the cosecant is the reciprocal of the sine. Also recall that the cosine graph is a sinusiodal graph with x-intercepts at x = n*pi/2, maximun value of 1 at x = 2n*pi and minimum value of -1 at x = (2n - 1)pi. The secant graph is of the shape of parabolas with vertices at the maximum/minimum values of the cosine graph and asymptotes at the x-intercepts of the cosine graph.
Similarly, recall that the sine graph is a sinusiodal graph with x-intercepts at x = 2n*pi, maximun value of 1 at x = pi/2 + 2n*pi and minimum value of -1 at x = -pi/2 + 2n*pi. The cosecant graph is of the shape of parabolas with vertices at the maximum/minimum values of the sine graph and asymptotes at the x-intercepts of the sine graph.
#graphtrig
#trigonometry #graphtrigfunctions

Views: 37174
Brian McLogan

Learn how to solve a system of equations by graphing. To solve a system of equations means to obtain a common x-value and a common y-value that makes the each of the equation in the system true. To solve a system of equations by graphing means to obtain the point of intersection (if any) of the graphs of each of the equation that make up the system. To graph a linear equation, we set the equation in the slope-intercept form and then graph the intercept and obtain the line using the slope.
The point of intersection of the graphs of the equations that make up the system is the solution to the system of equations.
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#systemsofequations #graphsystemsofequations

Views: 178548
Brian McLogan

Learn how to write the equation of a polynomial when given complex zeros. Recall that a polynomial is an expression of the form ax^n + bx^(n-1) + . . . + k, where a, b, and k are constants and the exponents are positive integers. The zeros of a polynomial are the values of x for which the value of the polynomial is zero. Also recall that when a complex number is a zero to a polynomial, the conjugate of the complex number will also be a zero to the polynomial.
To write the equation of a polynomial, we write the given zeros in factor form and expand the product of the factors. Thus, given a, b, . . . as zeros to a polynomial, we write the equation of the polynomial by expanding the factors (x - a)(x - b) . . . = 0
#polynomials #writepolynomial

Views: 30554
Brian McLogan