The slope characterizes the inclination angle to the horizontal axis (angular coefficient is numerically equal to the tangent of the angle). The slope is present in the equation of a straight line and is used in the mathematical analysis of the curves, where the slope is always equal to the derivative of the function. To better understand the slope, imagine that it affects the speed of function changing, that is, the greater the value of the slope, the greater the value of the function (with the same value of the independent variable).

The coefficient k in the line formula y = kx + b on the coordinate plane is numerically equal to the tangent of the angle, which constitutes the smallest turn of the Ox axis to the Oy axis, between the positive direction of the x-axis and a straight line.

The tangent of the angle can be calculated as the ratio of the opposite leg to an adjacent leg. K is always equal to DeltaY/DeltaX, i.e., a direct derivative of the equation with respect to x.

The slope does not exist (goes to infinity) for straight lines that are parallel to the axis Oy.

If the slope coefficient k is of a positive value and the coefficient of shift b is equal to zero, the line will be in the first and third quadrants (where x and y are simultaneously positive and negative). In addition, large values of the angular coefficient k will correspond to a steeper straight line, and smaller values will correspond to the a more gentle line.

Lines y = k_1x + b_1 and y = k_2x + b_2 are perpendicular if k_1k_2 = -1, and are parallel with k_1 = k_2.

The first slope formula is calculation of the slope of the straight line of equation.

Use the slope formula to find an inclination angle of the straight to the horizontal axis and the direction of this line. It is pretty easy to calculate the slope, if you are given the equation line. Remember that in any equation of the line:

- There are no exponents.
- There are only two variables, none of which is a fraction (for example, 1/2).
- The equation of the line is y = kx + b, where k and b are the numerical coefficients (e.g., 3, 10, -12, 4/3).

In order to find the slope you need to find the value of k (the coefficient of «x»). If the slope formula you have is y = kx + b, then to find the slope you just need to look at the number before the «x.» Note that k (slope) is always next to the independent variable (in this case «x»). If you are confused, the following examples will help you: y = kx + b, the slope = 2; y = 2 – x, the slope = -1; y = 3/8x – 10, the slope = 3.8/

If the slope formula you have is different from y = kx + b, isolate the dependent variable. In most cases, the dependent variable is referred to as «y», and in order to isolate it, one can perform actions of addition, subtraction, multiplication, and others. Remember that any mathematical operation to be performed on both sides of the equation (so as not to alter its original value). You need to bring an equation you have to the following slope formula: y = kx + b. For example, you need to find the slope of the equation 2y – 3 = 8x + 7. This equation should be reduced to the slope formula y = kx + b: 2y – 3(+3) = 8x + 7(+3); 2y = 8x + 10; 2y/2 = (8x + 10)/2; y = 4x + 5. The slope = k = 4.

The second slope formula is calculation of the slope by two points.

In order to calculate the slope, use the graph and two points. If you are given a graph of the function by itself (without slope formula), you can still find the slope. To do this, you need the coordinates of any two points on this chart. These coordinates will be substituted into the slope formula: (y2 – y1)/(x2 – x1). To avoid errors in the calculation of the slope, remember the following:

- If the graph increases, the slope is positive.
- If the graph is decreasing, then the slope is negative.
- The higher the value of the slope, the steeper the graph (and vice versa).
- The slope of the line, which is parallel to the x-axis, is equal to 0.
- The slope of the line, which is parallel to the y-axis, does not exist (it is infinite).

Now find the coordinates of two points. Mark any two points on the graph and find their coordinates (x, y). For example, on the graph there are points A (2,4) and B (6,6).

In a pair of coordinates the first number corresponds to the value of «x», and the second number corresponds to the value of «y». For each value of «x» there is a certain value «y».

Equate x1, y1, x2, y2 to the corresponding values. In our example equate them with the points A (2,4) and B (6,6): x1: 2, y1: 4, x2: 6, y2: 6.

Substitute the values you have found in the slope formula to calculate the slope. To find the slope, you need to use the coordinates of two points and the following slope formula: (y2 – y1)/(x2 – x1). Now put the coordinates of two points in the slope formula: Two points: A (2,4) and B (6,6). Substitute the coordinates of points in the slope formula: (6 - 4)/(6 -2). Simplify it to get a definitive answer: 2/4 = 1/2 = the angular coefficient.

The slope is the ratio of change in coordinates «y» (of two points) to the change of the coordinates of «x» (of two points). Changing coordinates is the difference between the values of the corresponding coordinates of the first and second points.

The standard slope formula for calculating the angular coefficient: k = (y2 -y1)/(x2 -x1). But it also can be as follows: k = Δy / Δx, where Δ - is the Greek letter «delta» that indicates the difference in math. That is, Δx = x2 - x1, and Δy = y2 -y1.

The third slope formula is the use of differential calculus to calculate the slope.

For this method you need to learn how to take derivatives of the functions. The derivative characterizes the rate of change of the function at a particular point on the graph of the function. In this case, the graph can be either straight or curved line. That is, the derivative characterizes the rate of change of the function at a particular time. Remember the general rules that are used for taking the derivatives, and only then proceed to the next step.

Learn to distinguish the problems, in which you need to calculate the slope using the derivative function. The tasks don’t always ask to find the slope or derivative of the function. For example, you may be asked to find the rate of change of the function at the point (x, y). Also, you may be asked to find the slope of the tangent at the point A (x, y). In both cases it is necessary to take the derivative of the function.

Let’s study that using the example: find the angular coefficient of the function f(x) = 2x2 + 6x at point A (4.2). Often the derivative is referred to as the f’(x), y’ or dy/dx.

Take the derivative of the function you have. It is not necessary to build a graph here – you will need only the equation function. In our example, take the derivative of the function f(x) = 2x2 + 6x. Remember to take the derivative according to the methods set out in this article: derivative: f’(x), = 4x + 6.

Now in the found derivative substitute the coordinates of the point you have to calculate the slope. The derivative of the function is the slope at a certain point. In other words, f’(x) is the slope of the function (x, f (x)) at any point. In our example: find the angular coefficient of the function f(x) = 2x2 + 6x at point A (4.2). The derivative of the function: f’(x) = 4x + 6. Substitute the value of «x» of the point: f’(x) = 4 (4) 6. Find the slope: the angular coefficient of the function f(x) = 2x2 + 6x at point A (4.2) is equal to 22.

If possible, check the answer on the chart. Note that the slope can’t be calculated at any point. Differential calculus considers the complex graphics and complex functions, where the slope can be calculated not at any point, but in some cases not all points lie in the graphs. If possible, use a graphing calculator to verify the correct calculation of the slope of the function. Otherwise draw a tangent to the graph at the point you have and think, whether the value of the slope you found corresponds to the value on the chart.

The tangent will have the same angular coefficient as the graph of a function at a certain point. To draw a tangent at this point, move to the right / left on the X axis (in our example, for 22 values to the right), and then up for one unit on axis Y. Mark the point, and then connect it to the point you have. In our example, connect the points with coordinates (4,2) and (26,3).

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