The slope of a line on the coordinate plane basically tells you how steep the line is. If you know the rise and run of a line, you can calculate its slope using the slope formula. Just make sure that you plug your numbers into the right places in the formula. Compare the following list with the second figure, which shows you that the slope of a line increases as the line gets steeper and steeper:.
The lines you see in the second figure have positive slopes except for the horizontal and vertical lines. So what about lines with negative slopes? Actually, there are a couple of ways to distinguish the two types of slopes:.
Mark Ryan is the founder and owner of The Math Center in the Chicago area, where he provides tutoring in all math subjects as well as test preparation. Understanding Line Slopes and the Slope Formula. Slope is the ratio of the rise to the run. The slope tells you how steep a line is.
A negative slope goes up to the left and down to the right. About the Book Author Mark Ryan is the founder and owner of The Math Center in the Chicago area, where he provides tutoring in all math subjects as well as test preparation.Many times in the study of statistics it is important to make connections between different topics.
We will see an example of this in which the slope of the regression line is directly related to the correlation coefficient.
Since these concepts both involve straight lines, it is only natural to ask the question, "How are the correlation coefficient and least square line related? It is important to remember the details pertaining to the correlation coefficient, which is denoted by r.
This statistic is used when we have paired quantitative data. From a scatterplot of paired datawe can look for trends in the overall distribution of data. Some paired data exhibits a linear or straight-line pattern. But in practice, the data never falls exactly along a straight line. Several people looking at the same scatterplot of paired data would disagree on how close it was to showing an overall linear trend. After all, our criteria for this may be somewhat subjective.
The scale that we use could also affect our perception of the data. For these reasons and more we need some kind of objective measure to tell how close our paired data is to being linear. The correlation coefficient achieves this for us. The last two items in the above list point us toward the slope of the least squares line of best fit. Recall that the slope of a line is a measurement of how many units it goes up or down for every unit we move to the right.
Sometimes this is stated as the rise of the line divided by the run, or the change in y values divided by the change in x values. In general, straight lines have slopes that are positive, negative, or zero. If we were to examine our least-square regression lines and compare the corresponding values of rwe would notice that every time our data has a negative correlation coefficientthe slope of the regression line is negative.
Similarly, for every time that we have a positive correlation coefficient, the slope of the regression line is positive. It should be evident from this observation that there is definitely a connection between the sign of the correlation coefficient and the slope of the least squares line. It remains to explain why this is true. The reason for the connection between the value of r and the slope of the least squares line has to do with the formula that gives us the slope of this line.
For paired data x,y we denote the standard deviation of the x data by s x and the standard deviation of the y data by s y. The calculation of a standard deviation involves taking the positive square root of a nonnegative number. As a result, both standard deviations in the formula for the slope must be nonnegative. If we assume that there is some variation in our data, we will be able to disregard the possibility that either of these standard deviations is zero.
Therefore the sign of the correlation coefficient will be the same as the sign of the slope of the regression line. Share Flipboard Email. Courtney Taylor. Professor of Mathematics. Courtney K.May 29, References. This article was co-authored by our trained team of editors and researchers who validated it for accuracy and comprehensiveness. This article has been viewedtimes. Learn more The slope of a line is a measure of how fast it is changing.Introduction to point-slope form - Algebra I - Khan Academy
This can be for a straight line -- where the slope tells you exactly how far up positive slope or down negative slope a line goes while it goes how far across. Slope can also be used for a line tangent to a curve. Or, it can be for a curved line when doing Calculus, where slope is also known as the "derivative" of a function.
Either way, think of slope simply as the "rate of change" of a graph: if you make the variable "x" bigger, at what rate does "y" change? That is a way to see slope as a cause and an effect event.
Aspects That Make Slope Formula Difficult For Many Students
Not quite! Looks like you may have calculated the equation correctly, but identified the wrong part of the solution as the slope. Instead, the correct answer will be the constant m. Choose another answer! Read on for another quiz question. You can divide the entire equation by this constant to isolate y. Not exactly! Next, you should try isolating the variable y.
Guess again! While this is technically correct, you should always simplify a slope to its simplest form. Pick another answer! Looks like you may have plugged the points into the point-slope formula incorrectly. It seems that you might have incorrectly applied the point-slope formula for this one. Plug the x of point 2, 4 into the derivative for a slope of You may have gotten this answer by incorrectly plugging the x value into the function before finding its derivative.
Remember, before you can plug in the x value, you must find a derivative of the function. It looks like you may have plugged the wrong value of point 2, 4 into a derivative for the line. Remember, you should plug the x value into the derivative, not the y value. That would be 2. Try again!
You might have gotten this answer by incorrectly plugging the y value into the function and trying to solve.To find the slope of a line we need two coordinates on the line. Any two coordinates will suffice. We are basically measuring the amount of change of the y-coordinate, often known as the risedivided by the change of the x-coordinate, known the the run. The calculations in finding the slope are simple and involves nothing more than basic subtraction and division.
Notice that the slope of a line is easily calculated by hand using small, whole number coordinates. The formula becomes increasingly useful as the coordinates take on larger values or decimal values.
It is worth mentioning that any horizontal line has a gradient of zero because a horizontal line has the same y-coordinates. This will result in a zero in the numerator of the slope formula. On the other hand, a vertical line will have an undefined slope since the x-coordinates will always be the same. This will result the division by zero error when using the formula. Just as slope can be calculated using the endpoints of a segment, the midpoint can also be calculated.
The midpoint is an important concept in geometry, particularly when inscribing a polygon inside another polygon with the its vertices touching the midpoint of the sides of the larger polygon.
This can be obtained using the midpoint calculator or by simply taking the average of each x-coordinates and the average of the y-coordinates to form a new coordinate. The slopes of lines are important in determining whether or not a triangle is a right triangle. If any two sides of a triangle have slopes that multiply to equal -1, then the triangle is a right triangle. The computations for this can be done by hand or by using the right triangle calculator.
You can also use the distance calculator to compute which side of a triangle is the longest, which helps determine which sides must form a right angle if the triangle is right. The sign in front of the gradient provided by the slope calculator indicates whether the line is increasing, decreasing, constant or undefined. If the graph of the line moves from lower left to upper right it is increasing and is therefore positive. If it decreases when moving from the upper left to lower right, then the gradient is negative.
The method for finding the slope from an equation will vary depending on the form of the equation in front of you. If the equation is not in this form, try to rearrange the equation.
To find the gradient of other polynomials, you will need to differentiate the function with respect to x. The rate of change of a graph is also its slopewhich are also the same as gradient. Rate of change can be found by dividing the change in the y vertical direction by the change in the x horizontal direction, if both numbers are in the same units, of course.
Rate of change is particularly useful if you want to predict the future of previous value of somethingas, by changing the x variable, the corresponding y value will be present and vice versa.
Slopes or gradients have a number of uses in everyday life. There are some obvious physical examples - every hill has a slope, and the steeper the hill, the greater its gradient. This can be useful if you are looking at a map and want to find the best hill to cycle down. You also probably sleep under a slope, a roof that is. The slope of a roof will change depending on the style and where you live. But, more importantly, if you ever want to know how something changes with time, you will end up plotting a graph with a slope.March 5, References.
This article was co-authored by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience.
She has taught math at the elementary, middle, high school, and college levels. There are 11 references cited in this article, which can be found at the bottom of the page. This article has been viewedtimes. The slope of a line measures how steep the line is. Grace Imson, MA. Our Expert Agrees: If you have the slope and one point, plug them into the equation of the line. Then, solve for b to find the y-intercept. To find the slope of a line from a graph, first choose 2 points along the line and write down the X and Y coordinates for each.
Next, find the rise by taking the difference between the 2 Y coordinates. If the line slopes up as it moves to the right, the rise will be positive. If it slopes down, the rise will be negative. Finally, find the slope by dividing the rise by the run. Keep reading for more tips, including how to find the Y-intercept using the slope and 1 point!
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What Is the Slope Formula?
Draw dots on the graph to represent these points, and note their coordinates. Remember when graphing points to list the x-coordinate first, then the y-coordinate. For example, you might choose the points -3, -2 and 5, 4. Determine the rise between the two points. To do this, you must compare the difference in y of the two points.
Begin with the first point, the point that is the farthest left on the graph, and count up until you reach the y-coordinate of the second point. The rise can be positive or negative; that is, you can count up or down to find it. If the line is moving down and to the right, the rise is negative.Do you have trouble remembering all of the algebra formulas, especially the slope formula?
SLOPE Function in Excel
If so, you're in luck! Here you'll find a quick reference sheet with all of the formulas for slope. The slope is calculated by counting the rise and then counting the run.
We then write the slope as a fraction. We use this definition when calculating slope or graphing slope. Graphing slope also leads us to a very popular method for graphing linear equations, slope intercept form. When a linear equation is written in slope intercept form, the slope of the line can easily be identified.
The slope is "m" or the coefficient of x in the equation. Often times a graph is not present, and we must calculate the slope when given two ordered pairs.
In this case we must use another special formula. This formula is commonly used to solve rate of change problems. Click here for detailed examples on using this formula. Keeping a reference sheet of formulas is a great way to study Algebra.
Slope is a very important concept to remember in Algebra, so make sure you add these formulas and definitions to your reference sheet or study guide. We have a new platform with updated videos and worksheets. Click here to login to our Learn Worlds platform. On this site, I recommend only two products that I use and love. One is Mathway and the other is Magoosh.
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Point-Slope Equation of a Line
Affiliate Products Let Us Know How we are doing!To calculate the slope of a line you need only two points from that line, x 1y 1 and x 2y 2. The equation used to calculate the slope from two points is: On a graph, this can be represented as:. There are three steps in calculating the slope of a straight line when you are not given its equation. Let's say that points 15, 8 and 10, 7 are on a straight line. What is the slope of this line?
It doesn't matter which we choose, so let's take 15, 8 to be x 2y 2. Let's take the point 10, 7 to be the point x 1y 1. Once we've completed step 2, we are ready to calculate the slope using the equation for a slope:.
We said that it really doesn't matter which point we choose as x 1y 1 and the which to be x 2y 2. Let's show that this is true. Take the same two points 15, 8 and 10, 7but this time we will calculate the slope using 15, 8 as x 1y 1 and 10, 7 as the point x 2y 2. Then substitute these into the equation for slope:. Often you will not be given the two points, but will need to identify two points from a graph.
In this case the process is the same, the first step being to identify the points from the graph. Below is an example that begins with a graph. Example What is the slope of the line given in the graph? The slope of this line is 2. Notice that the line with the greater slope is the steeper of the two.
The greater the slope, the steeper the line. Keep in mind, you can only make this comparison between lines on a graph if: 1 both lines are drawn on the same set of axes, or 2 lines are drawn on different graphs i. You are now ready to try a practice problem. If you have already completed the first practice problem for this unit you may wish to try the additional practice. What is the slope of the line given in the graph?