Video: Slope of a Line from a Graph or Table

Video Transcript

In this video, we will learn how to find the slope of a line using graphs and tables. We will begin by recalling some key facts of linear functions. The graph of any linear function is a straight line. And the equation of any linear function is written in the form 𝑦 equals 𝑚𝑥 plus 𝑏. The letters 𝑚 and 𝑏 are constants, where 𝑚 represents the slope or gradient of the line. 𝑏 represents the 𝑦-intercept, the point at which our line crosses the 𝑦-axis. This is sometimes written as 𝑦 equals 𝑚𝑥 plus 𝑐 instead of 𝑏.

The value of 𝑚 will be positive if our straight line slopes up from left to right. 𝑚 will be negative if our line slopes down from left to right. The absolute value of 𝑚 determines how steep the slope is and its sign gives the direction of the slope. For example, the equation 𝑦 equals three 𝑥 plus four will be steeper than the equation 𝑦 equals two 𝑥 minus seven. This is because the value of 𝑚 is greater. As 𝑚 represents the slope, it follows that the value of 𝑚 is the rate of vertical change in the 𝑦-coordinates to the horizontal change in the 𝑥-coordinates between any two points.

This can be written using the following formula. 𝑚 is equal to 𝑦 two minus 𝑦 one divided by 𝑥 two minus 𝑥 one, where two points on the line 𝐴 and 𝐵 have coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two. This is often referred to as the change in 𝑦 over the change in 𝑥 or the rise over the run. We will now look at how we can apply this to find the slope of a linear function given its graph.

What is the slope of the function represented by the given figure?

We know that any straight line graph must be a linear function written in the form 𝑦 equals 𝑚𝑥 plus 𝑏, where the value of 𝑚 is the slope or gradient of the function. The value of the slope 𝑚 can be calculated using the following formula. 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one, where 𝐴 and 𝐵 are two coordinates on the line with coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two. We begin by selecting any two points on our straight line. If possible, it is often useful to choose points where the line crosses the 𝑥- and 𝑦-axis. In this case, 𝐴 has coordinates zero, 10 and 𝐵 has coordinates five, zero.

At this stage, it often helps to create a right-angled triangle on our graph. This will help us calculate the change in 𝑦 and the change in 𝑥, otherwise known as the rise and the run. Substituting our 𝑦-coordinates into the formula gives us zero minus 10. Substituting in our 𝑥-coordinates gives us five minus zero. It doesn’t matter which point is 𝑥 one, 𝑦 one and which is 𝑥 two, 𝑦 two. But we must be consistent in our order. Zero minus 10 is equal to negative 10. Five minus zero is equal to five. Negative 10 divided by five is equal to negative two. This means that the slope of the function represented on the graph is negative two.

We can check this on the graph by considering the rise and the run. The rise is negative 10, as the 𝑦-coordinate drops from 10 to zero. The run is five, as the 𝑥-coordinate goes from zero to five. Once again, we have negative 10 divided by five. An important check is that any line that goes up from left to right will have a positive slope. Any line that goes down from left to right will have a negative slope. As our line goes downwards from left to right, a negative answer is sensible.

We will now look at a second graph question.

Calculate the slope of the line in the graph.

We know that any straight line is a linear function that can be written in the form 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope or gradient of the line. The value of 𝑚 can be calculated using the formula 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. This is the change in 𝑦-coordinates over the change in 𝑥-coordinates, otherwise known as the rise over the run. We begin by selecting any two points on the line 𝐴 and 𝐵 with coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two. Whilst it doesn’t matter which two points we choose, it is sensible to pick those with integer coordinates where possible.

In this question, we will choose the two points shown on the graph. Point 𝐴 has coordinates zero, one and point 𝐵 has coordinates two, seven. At this point, it is worth drawing a right-angled triangle on the graph to show the rise and the run. The rise in this case is equal to six, as the change in 𝑦-coordinates is six. The run is equal to two. This means that we would expect the slope to be six divided by two, which is three.

We can check this by substituting our coordinates into the formula. The two 𝑦-coordinates were seven and one. And the corresponding 𝑥-coordinates were two and zero. This simplifies to six over two, which once again gives us an answer of three. The slope of the line in the graph is three. It is worth recalling that any line that slopes upwards from left to right will have a positive slope. As three is positive, this suggests that our answer is correct.

We will now look at a question that involves finding the slope of a linear function from a table.

What is the slope of the linear function represented by the given table?

We know that the equation of any linear function is written in the form 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope or gradient of the function and 𝑏 is the 𝑦-intercept. We can calculate the value of the slope 𝑚 using the following formula, 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. This is the change in 𝑦-coordinates over the change in 𝑥-coordinates, where any two points 𝐴 and 𝐵 have coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two, respectively.

In our table, we have three coordinates, firstly, zero, four. Our second coordinate has an 𝑥-value of two and a 𝑦-value of 10. Our third coordinate, which we will call 𝐶, is four, 16. We can select any two of these three coordinates. In this question, we will begin by considering point 𝐴 and point 𝐵. The 𝑦-coordinates of these two points are 10 and four. The corresponding 𝑥-coordinates are two and zero. The slope 𝑚 is equal to 10 minus four over two minus zero. This simplifies to six over two, giving us a final answer of a slope of three.

We can check this answer by selecting a different two points, in this case point 𝐴 and point 𝐶. This time the slope is equal to 16 minus four over four minus zero. 12 divided by four is also equal to three. We would get the same answer if we use the points 𝐵 and 𝐶. The slope of the linear function represented by the table is three.

We could also calculate this answer just by looking at the table. The change in 𝑥-values between the first and second point is plus two. The change in the 𝑦-values between the first two points is plus six. As the slope is equal to the change in 𝑦-values divided by the change in 𝑥-values, this also gives us an answer of three. For each single unit the 𝑥-value increases, the 𝑦-value will increase by three units.

Our next question will include a graph in a real-world context.

The graph shows the distance Amelia traveled over her two-hour bike ride. Which of the following is true? A) She traveled at a constant speed of four miles per hour for the last hour. B) She traveled at a constant speed of 10 miles per hour for the entire ride. C) She traveled at a constant speed of eight miles per hour for the last hour. Or D) she traveled at a constant speed of seven miles per hour for the entire ride.

We can see from the graph that the 𝑥-axis represents the time in hours and the 𝑦-axis represents the distance in miles. The speed or velocity in any distance-time graph can be calculated by dividing the change in distance between any two points by the change in time. If the graph is a straight line for the entire journey, then they will be traveling at a constant speed. We can see from the graph that three parts of the journey have different slopes or gradients. This means that, during these three parts, Amelia will be traveling at different speeds.

We can therefore rule out options B and D, as these stated that she traveled at a constant speed for the entire ride. This is not the case as she will have traveled at three different speeds. Both of the other statements relate to the last hour of Amelia’s journey. This occurs between the two points 𝐴 and 𝐵 on the graph. We can calculate the slope between any two points on a graph by using the following formula, 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. This is the change in 𝑦-coordinates over the change in 𝑥-coordinates, in this case the change in the distance over the change in the time.

Point 𝐴 has coordinates one, 10 and point 𝐵 has coordinates two, 14. The 𝑦-coordinates or distances here are 14 and 10. The corresponding 𝑥-coordinates are two and one. 14 minus 10 is equal to four and two minus one is equal to one. This means that the slope of the line between points 𝐴 and 𝐵 is four. We could also have worked this out by drawing a right-angled triangle on the graph. We can see here that the distance has risen from 10 to 14. And the time has gone from one hour to two hours. Four divided by one is equal to four. So once again, the slope equals four.

As the slope in a distance-time graph is equal to the speed, we can conclude that the speed in the last hour was four miles per hour. This rules out option C and therefore option A is correct. Amelia traveled at a constant speed of four miles per hour for the last hour.

We will now recap some of the key points from this video. The graph of any linear function is a straight line. A linear function has a constant rate of change, which means that the difference in the 𝑦-coordinates of any two points on the straight line is proportional to the difference in their 𝑥-coordinates. This rate of change is the slope of the line. The equation of a line is generally written in the form 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope or gradient of the line and 𝑏 is the 𝑦-intercept. This is the point where the line crosses the 𝑦-axis.

Finally, the slope of a line 𝑚 is the rate of the vertical change to the horizontal change between two points. For two points 𝐴 𝑥 one, 𝑦 one and 𝐵 𝑥 two, 𝑦 two lying on a line, the slope is 𝑚, which is equal to 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. If this number is positive, our line will slope upwards from left to right. And if it is negative, it will slope downwards from left to right.