- Linear Functions
- Quadratic Functions
- Exponential Functions
- Logarithmic Functions
- Rational Functions
- Absolute Value Functions
- Polynomial Functions
- Trigonometry Functions
- Exponential Functions
- Hyperbolic Functions
Linear functions are one of the most basic algebraic functions and are essential to many mathematical and practical applications. A straight line represents the graph of a linear function, which has a constant rate of change. A linear function has the general form y = mx + b, where m denotes the line's slope and b is the y-intercept.
The ratio of the change in y over the change in x is known as the slope of a linear function, and it indicates how quickly the function is changing. The slope might be positive, negative, zero, or undefined. It is defined as the ratio of vertical change to horizontal change. The line will slant upward to the right if the slope is positive, and downward to the right if the slope is negative. The line will be vertical if the slope is undefined and horizontal if the slope is zero.
The point where a line meets the y-axis is known as the y-intercept of a linear function. The value of y at x=0 is known as the y-intercept. Because it provides us with a starting point on the line, the y-intercept is crucial for charting linear functions.
Many real-world circumstances, such as the relationship between time and distance or the link between cost and quantity, can be modeled using linear functions. A linear function, for instance, can be used to calculate the distance an automobile will go at a given speed. d = rt, where r is the rate of speed and t is the time, is the equation for this function. In this instance, the car's starting position is represented by the y-intercept, and its starting speed is represented by the line's slope.
Calculating the slope and y-intercept of a line from two points is another significant application of linear functions. The equation m = (y2 - y1) / (x2 - x1), where m is the slope and (x1, y1) and (x2, y2) are the coordinates of the two points on the line, is used to determine the slope of a line given two points.
The slope-intercept version of the equation of a line is used to get the y-intercept of a line given two points: y - y1 = m(x - x1), where y1 is the y-coordinate of one of the line's points, m is the slope, and x1 is the line's x-coordinate. The y-intercept can then be determined by solving for y.
Algebraic functions that describe a parabola are called quadratic functions. In the formula y = ax2 + bx + c, where a, b, and c are constants, a quadratic function is defined. The parabola's shape is determined by the coefficient a, while its location on the coordinate plane is determined by the constants b and c.
You must know the vertex, y-intercept, and axis of symmetry to graph a quadratic function. The line dividing the parabola into two symmetrical halves is known as the axis of symmetry. The parabola's vertex is its highest or lowest point, and its y-intercept is the value of y at the point where x = 0.
Let's plot the function y = x2 - 2x + 1 as an illustration. This function's vertex is at (1, 0) and its symmetry axis is at x = 1. One is the y-intercept. These points can be plotted, and a parabola can be drawn to pass through them.
Exponential functions are algebraic functions used to model how a quantity changes over time. Y = abx, where a is the starting value, b is the growth or decay factor, x is the time variable, and y is the formula for an exponential function.
You must know the initial value and the growth or decay factor to graph an exponential function. The function denotes exponential growth if the growth or decay factor is larger than 1. It describes exponential decay if it is between 0 and 1.
Let's plot the function y = 2(3)x as an illustration. This function's initial value is 2, and the growth factor is 3. To create the function's graph, we can draw the points (0, 2), (1, 6), (2, 18), and so on and connect them with a straight line.
The inverse connection between exponential functions is described by logarithmic functions. The formula for a logarithmic function is y = logb(x), where b is the logarithm's base. The power to which the base b must be raised to acquire a number x is its logarithm.
The base and the vertical asymptote must be determined before you can graph a logarithmic function. The line that the graph approaches but never crosses is the vertical asymptote. The x-axis serves as the vertical asymptote for logarithmic functions.
Let's plot the function y = log2(x) as an illustration. This function has a base of 2, and the x-axis serves as its vertical asymptote. Plotting certain points on the function, like (0, 0), (1, 1), (4, 2), and so on, allows us to create a smooth curve that moves closer to the x-axis.
Rational functions describe the relationship between two polynomials. When y = p(x)/q(x), where p(x) and q(x) are polynomials, it is a rational function. A rational function's graph is a curve with potential asymptotes on both the horizontal and vertical axes.
Finding the horizontal and vertical asymptotes, intercepts, and critical points is necessary to graph a rational function. The horizontal asymptotes are the values of y that the function approaches as x becomes very large or very tiny, whereas the vertical asymptotes are the values of x that make the denominator zero.
Let's plot the function y = (x+1)/(x2-4) as an illustration. This function has x = 2 and x = -2 as its vertical asymptotes and y = 0 as its horizontal asymptote. By making y = 0 and solving for x, we may also determine the x-intercept, which results in x = -1. These points can then be plotted, and a curve that approaches the asymptotes can be created.
Algebraic functions known as absolute value functions quantify the separation between a given number and zero on the number line. Y = |x|, where |x| represents the absolute value of x, is the formula for an absolute value function. A V-shaped curve represents the graph of an absolute value function.
You must know the vertex and the slope of the V-shaped curve to graph an absolute value function. The function's vertex is located at its origin, and its arms' slope is 1.
Let's plot the function y = |x| as an example. This function's origin serves as its vertex, and its arms' slope is 1. Then, using a V-shaped curve, we may link certain points, like (-2, 2), (-1, 1), (0, 0), (1, 1), and (2, 2), that have been plotted.
Polynomial functions explain the relationship between a polynomial's terms. The formula for a polynomial function is y = a0 + a1x + a2x2 +... + an x n, where a0, a1, a2,..., an are constants and n is a positive integer.
You must know the polynomial's degree, the leading coefficient, and the intercepts to graph it. The leading coefficient is the coefficient of the term with the largest power of x, and the degree of the polynomial is the highest power of x in the function.
Let's plot the function y = x 3 - 3x 2 + 2x + 1 as an illustration. This polynomial has a degree of 3, and its leading coefficient is 1. By setting y = 0 and solving for x, we can determine the x-intercepts, which results in x = -1, x = 1, and x = 2. These points can then be plotted, and a curve can be drawn to pass through them.
Trigonometric functions explain how a right triangle's angles and sides relate to one another. Sine, cosine, tangent, cosecant, secant, and cotangent are the six trigonometric functions. These formulas can alternatively be expressed as right triangle side ratios.
It is necessary to know the period, amplitude, phase shift, and intercepts to graph a trigonometric function. The amplitude of a trigonometric function is the distance from the center line to the peak or trough, whereas the period is the separation between two successive peaks or troughs. The intercepts are the locations where the function crosses the x- or y-axis, whereas the phase shift is the function's horizontal shift.
Let's plot the function y = 2sin(2x - pi/2) as an illustration. This function has a pi period, a 2 amplitude, and a pi/4 phase shift. Setting y = 0 and solving for x results in x = pi/8, 3pi/8, 5pi/8, and 7pi/8, from which we may deduce the x-intercepts. These points can then be plotted, and a curve can be drawn to pass through them.
Exponential functions explain the link between a fixed base and a variable exponent. The formula for an exponential function is y = abx, where an is a constant and b is the function's base. A curve that either climbs or drops exponentially makes up the graph of an exponential function.
The base and intercepts of an exponential function must be determined before you can graph it. Whether a function increases or declines exponentially depends on its basis. The places where the function crosses either the x-axis or the y-axis are known as the intercepts.
Let's plot the function y = 2(3)x as an illustration. This function's y-intercept is 2 and its base is 3. Setting y = 0 and solving for x results in x = log3(1/2), the x-intercept. Then, using a visualization of these points, we can create an exponential growth curve.
Algebraic functions known as hyperbolic functions explain how the unit circle and the hyperbola relate to one another. The six hyperbolic functions are the hyperbolic sine, the hyperbolic cosine, the hyperbolic tangent, the hyperbolic cosecant, and the hyperbolic secant. These formulas can alternatively be expressed as right triangle side ratios.
You must determine the period, amplitude, phase shift, and intercepts to graph a hyperbolic function. The amplitude of a hyperbolic function is the distance from the center line to the peak or trough, whereas the period is the separation between two successive peaks or troughs. The intercepts are the locations where the function crosses the x- or y-axis, whereas the phase shift is the function's horizontal shift.
Let's plot the function y = cosh(x) as an illustration. This function has an infinite period, a single amplitude, and a zero phase shift. By setting x = 0, which results in y = 1, we may get the y-intercept. Then, using the function, we can plot a few points, like (-2, 3.76), (-1, 1.54), (0, 1), (1, 1.54), and (2, 3.76), and then create a smooth curve that goes across those places.
Mathematical algebraic functions are an integral aspect of many disciplines, including science, engineering, finance, and economics. Understanding and solving challenging mathematics problems will be much easier if you are familiar with the top 10 algebraic functions covered in this article.