Gradient and Intercept
In mathematics, the gradient and intercept are key components of a linear equation in the form y = mx + c, where m represents the gradient and c represents the y-intercept. The gradient indicates the steepness of the line, while the intercept represents the point where the line crosses the y-axis. These parameters are fundamental in understanding and graphing linear functions.
4 Key excerpts on "Gradient and Intercept"
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...The slope tells us the rate of change, so we know how the x and y values increase or decrease together. For example, if you start at point (1, 2) and move using a slope of 3, you can complete the table. Now you can use the table of values to graph the line. You can also use the equation. DEFINITION Slope-intercept form The equation of a straight line in the form y = mx + b, where m is the slope and b is the y -coordinate of the point where the line intercepts the y -axis. This requires the equation to be in y = form. If so, it is that simple. If an equation is linear (first-degree x), then you can use slope-intercept form to find the slope. Remember, slope is a ratio, so you are going to want to write the slope as a fraction. • Write the equation in y = form (y = mx + b). • Find the coefficient of x. • This number is the slope. • If there is no x, there are zero x. • Find the constant. • This number is the y -intercept. • If there is no constant, it is zero. Knowing the y -intercept gives you a point (0, b). You can choose to continue to make a chart starting with the y -intercept as your first number, or go straight to the graph. The y -intercept is where you begin and the slope is how you move. Remember, m for move and b for begin. The slope is 2. Think 2 up / 1 right. The y -intercept is –3. Begin at (0, –3) and move 2 up, 1 right. Usually you will want to make at least three points before you draw the line. Label your line with the equation. The slope is 3. Think 3 up / 1 right. The y -intercept is 1. Begin at (0, 1) and move 3 up, 1 right. Usually you will want to make at least three points before you draw the line. Label your line with the equation. The slope is. Think 1 up / 2 right. The y -intercept is 4. Begin at (0, 4) and move 1 up, 2 right. Usually you will want to make at least three points before you draw the line...
eBook - ePub- Jim Fowler, Lou Cohen, Philip Jarvis(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
...In the general case, the gradient is equal to an increment in y divided by a corresponding increment in x (Fig. 15.2). Knowing the value of b we may use it to calculate the height gained from a given distance moved horizontally, thus: Vertical height gained = gradient × horizontal distance travelled In the general case, this may be expressed as: Fig. 15.1 A regression line. If we wish to know the actual final height rather than just the height gained we must know the height of our starting point above some reference zero, say sea level. If this height is symbolized by a units, the final height will be: Final height = height of starting point + (gradient × horizontal distance travelled) or, in the general case (see Fig. 15.3), The equation, y = a + bx, is known as the equation of a straight line or the rectilinear equation. Regression analysis is concerned with solving the values of a and b in the equation from a set of bivariate sample data. We may then accurately fit a line to a scattergram and estimate the value of one variable from a measurement of the other. The quantities a and b are both regression coefficients. In common usage, however, the term ‘regression coefficient’ is taken to mean the slope of the regression line, b. Fig. 15.2 Gradients of slopes: gradient = y units/ x units. Fig. 15.3 Final height of car above sea level = a + bx where b is the gradient. 15.3 Dependent and independent variables To this point we have not indicated which of two variables should be placed on the y (vertical) axis and which on the x (horizontal) axis. There is a convention which gives us a guideline in this respect. In many pairs of variables it is possible to discern, unambiguously, that one of the variables is dependent on the other. For example, the number of warblers breeding in a county might depend on the length of hedgerow available...
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...However, to make a more quantitative statement about the slope we need specific information about a pair of points that both lie on the line. Inspection of EQ5.1 reveals that f = 32 when c = 0 and that f = 212 when c = 100. If we define the slope of the line as the increase in the value of the function f for a given increase in the value of the variable c, then: Slope = f (100) − f (0) 100 − 0 = 212 − 32 100 = 1.8 = 9 5. Interestingly, the value of the slope determined here is the same as the coefficient of the c term in EQ5.1. More generally, we can try the same approach using some basic coordinate geometry and the generic formula for a straight line: y (x) = A + B x. (EQ5.2) Choosing any two pairs of coordinates that lie on this line, (x 1, y 1) and (x 2, y 2), we can determine the slope (Figure 5.6). However, the coordinates of both of these points must satisfy EQ5.2, which means that y 1 = A + Bx 1 and y 2 = A + Bx 2. Using this information to calculate the slope, namely the rate of increase of y with respect to an increase in x, we get: Slope = y 2 − y 1 x 2 − x 1 = (A + B x 2) − (A + B x 1) x 2 − x 1 = B (x 2 − x 1) x 2 − x 1 = B. This result proves that when the equation of a straight line is written in the form y = A + Bx, the gradient is always represented by the value of the constant B. Figure 5.6 The gradient (or slope) of a straight line is equal to. (y 2 – y 1)/(x 2 – x 1), the increase in y values divided by the increase in x values. Figure 5.7 The position of an apple at 0, 20, 40, and 60 ms after it begins its descent toward the head of Isaac Newton. Figure 5.8 A plot of y against x for y (x) = 0.05 x 2...
eBook - ePub- Steven Strom, Kurt Nathan, Jake Woland(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
...The terms grade and gradient are commonly used synonymously with slope. A generalized definition of slope, then, is the vertical change in elevation (fall or rise in feet or meters) in a horizontal distance or S = DE/L, where S is the slope and DE is the difference in elevation between the end points of a line of which the horizontal or map distance is L (Figure 4.11). To express S as a percentage, multiply the value by 100. Figure 4.11. Diagram of the slope formula. One problem that commonly arises is that L is measured horizontally rather than along the slope. To reinforce this point, it should be remembered from surveying that all map distances are measured horizontally, not parallel to the surface of sloping ground. With the slope formula, three basic computations may be accomplished: Knowing the elevations at two points and the distance between the points, slope S can be calculated. Knowing the difference in elevation between two points and the percentage of slope, the horizontal distance L can be calculated. Knowing the percentage of slope and the horizontal distance, the difference in elevation DE can be calculated. The following sample problems illustrate the three basic applications of the slope formula. Example 4.5 Two spot elevations are located 120 ft. apart (measured horizontally). One spot elevation is 44.37 ft., while the other is 47.81 ft. (Figure 4.12)...
