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- https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_(Arnold)/03%3A_Linear_Functions/3.01%3A_Linear_ModelsIf we assume that the distance is a continuous function of time, a legitimate assumption due to the fact that the distance is increasing continuously at a constant rate of 4 feet per second, then we c...If we assume that the distance is a continuous function of time, a legitimate assumption due to the fact that the distance is increasing continuously at a constant rate of 4 feet per second, then we can replace the discrete set of data points in Figure \PageIndex5(a) with the line shown in Figure \PageIndex5(b).
- https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_(Arnold)/09%3A_Radical_Functions/9.06%3A_The_Pythagorean_TheoremPythagoras was a Greek mathematician and philosopher, born on the island of Samos (ca. 582 BC). He founded a number of schools, one in particular in a town in southern Italy called Crotone, whose memb...Pythagoras was a Greek mathematician and philosopher, born on the island of Samos (ca. 582 BC). He founded a number of schools, one in particular in a town in southern Italy called Crotone, whose members eventually became known as the Pythagoreans. Today, nothing is known of Pythagoras’s writings, perhaps due to the secrecy and silence of the Pythagorean society. However, one of the most famous theorems in all of mathematics does bear his name, the Pythagorean Theorem.
- https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_(Arnold)/07%3A_Rational_Functions/7.03%3A_Graphing_Rational_FunctionsWe’ve seen that the denominator of a rational function is never allowed to equal zero; division by zero is not defined. So, with rational functions, there are special values of the independent variabl...We’ve seen that the denominator of a rational function is never allowed to equal zero; division by zero is not defined. So, with rational functions, there are special values of the independent variable that are of particular importance. Now, it comes as no surprise that near values that make the denominator zero, rational functions exhibit special behavior, but here, we will also see that values that make the numerator zero sometimes create additional special behavior in rational functions.
- https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_(Arnold)/03%3A_Linear_Functions/3.02%3A_SlopeIn the previous section on Linear Models, we saw that if the dependent variable was changing at a constant rate with respect to the independent variable, then the graph was a line. You may have also l...In the previous section on Linear Models, we saw that if the dependent variable was changing at a constant rate with respect to the independent variable, then the graph was a line. You may have also learned that higher rates led to steeper lines (lines that rose more quickly) and lower rates led to lines that were less steep. In this section, we will connect the intuitive concept of rate developed in the previous section with a formal definition of the slope of a line.
- https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_(Arnold)/05%3A_Quadratic_Functions/5.04%3A_The_Quadratic_FormulaThe equation ax²+bx+c=0 is called a quadratic equation. Previously, we solved equations of this type by factoring and using the zero product property. It is not always possible to factor the trinomial...The equation ax²+bx+c=0 is called a quadratic equation. Previously, we solved equations of this type by factoring and using the zero product property. It is not always possible to factor the trinomial on the left-hand side of the quadratic equation as a product of factors with integer coefficients and we’ll need another method to solve the quadratic equation; the purpose of this section is to develop a formula that will consistently provide solutions of the general quadratic equation.
- https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_(Arnold)/06%3A_Polynomial_Functions/6.02%3A_Zeros_of_PolynomialsIn the previous section we studied the end-behavior of polynomials. In this section, our focus shifts to the interior. There are two important areas of concentration: the local maxima and minima of th...In the previous section we studied the end-behavior of polynomials. In this section, our focus shifts to the interior. There are two important areas of concentration: the local maxima and minima of the polynomial, and the location of the x-intercepts or zeros of the polynomial. In this section we concentrate on finding the zeros of the polynomial.
- https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_(Arnold)/02%3A_Functions/2.02%3A_The_Graph_of_a_FunctionDescartes introduces his coordinate system, a method for representing points in the plane via pairs of real numbers. Indeed, the Cartesian plane of modern day is so named in honor of Rene Descartes, w...Descartes introduces his coordinate system, a method for representing points in the plane via pairs of real numbers. Indeed, the Cartesian plane of modern day is so named in honor of Rene Descartes, who some call the “Father of Modern Mathematics.” A Cartesian Coordinate System consists of a pair of axes, usually drawn at right angles to one another in the plane, one horizontal (labeled x) and one vertical (labeled y).
- https://math.libretexts.org/Courses/Queens_College/Preparing_for_Calculus_Bootcamp_(Gangaram)/06%3A_Day_6/6.01%3A_The_Pythagorean_TheoremPythagoras was a Greek mathematician and philosopher, born on the island of Samos (ca. 582 BC). He founded a number of schools, one in particular in a town in southern Italy called Crotone, whose memb...Pythagoras was a Greek mathematician and philosopher, born on the island of Samos (ca. 582 BC). He founded a number of schools, one in particular in a town in southern Italy called Crotone, whose members eventually became known as the Pythagoreans. Today, nothing is known of Pythagoras’s writings, perhaps due to the secrecy and silence of the Pythagorean society. However, one of the most famous theorems in all of mathematics does bear his name, the Pythagorean Theorem.
- https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_(Arnold)/06%3A_Polynomial_Functions/6.03%3A_Extrema_and_ModelsIn the last section, we used end-behavior and zeros to sketch the graph of a given polynomial. We also mentioned that it takes a semester of calculus to learn an analytic technique used to calculate t...In the last section, we used end-behavior and zeros to sketch the graph of a given polynomial. We also mentioned that it takes a semester of calculus to learn an analytic technique used to calculate the “turning points” of the polynomial. That said, we’ll still pursue the coordinates of the “turning points” in this section, but we will use the graphing calculator to assist us in this quest; and then we will use this technique with some applications.
- https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_(Arnold)/05%3A_Quadratic_Functions/5.03%3A_Zeros_of_the_QuadraticWhen drawing the graph of the parabola it is helpful to know where the graph of the parabola crosses the x-axis. That is the primary goal of this section, to find the zero crossings or x-intercepts of...When drawing the graph of the parabola it is helpful to know where the graph of the parabola crosses the x-axis. That is the primary goal of this section, to find the zero crossings or x-intercepts of the parabola.
- https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_(Arnold)/05%3A_Quadratic_Functions/5.05%3A_MotionIf a particle moves with uniform or constant acceleration, then it must behave according to certain standard laws of kinematics. In this section we will develop these laws of motion and apply them to ...If a particle moves with uniform or constant acceleration, then it must behave according to certain standard laws of kinematics. In this section we will develop these laws of motion and apply them to a number of interesting applications.
