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In this video, we will learn how to find the slope of a line using graphs and tables.
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We will begin by recalling some key facts of linear functions.
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The graph of any linear function is a straight line.
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And the equation of any linear function is written in the form π¦ equals ππ₯ plus π.
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The letters π and π are constants, where π represents the slope or gradient of the line.
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π represents the π¦-intercept, the point at which our line crosses the π¦-axis.
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This is sometimes written as π¦ equals ππ₯ plus π instead of π.
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The value of π will be positive if our straight line slopes up from left to right.
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π will be negative if our line slopes down from left to right.
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The absolute value of π determines how steep the slope is and its sign gives the direction of the slope.
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For example, the equation π¦ equals three π₯ plus four will be steeper than the equation π¦ equals two π₯ minus seven.
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This is because the value of π is greater.
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As π represents the slope, it follows that the value of π is the rate of vertical change in the π¦-coordinates to the horizontal change in the π₯-coordinates between any two points.
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This can be written using the following formula.
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π is equal to π¦ two minus π¦ one divided by π₯ two minus π₯ one, where two points on the line π΄ and π΅ have coordinates π₯ one, π¦ one and π₯ two, π¦ two.
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This is often referred to as the change in π¦ over the change in π₯ or the rise over the run.
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We will now look at how we can apply this to find the slope of a linear function given its graph.
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What is the slope of the function represented by the given figure?
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We know that any straight line graph must be a linear function written in the form π¦ equals ππ₯ plus π, where the value of π is the slope or gradient of the function.
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The value of the slope π can be calculated using the following formula.
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π¦ two minus π¦ one over π₯ two minus π₯ one, where π΄ and π΅ are two coordinates on the line with coordinates π₯ one, π¦ one and π₯ two, π¦ two.
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We begin by selecting any two points on our straight line.
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If possible, it is often useful to choose points where the line crosses the π₯- and π¦-axis.
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In this case, π΄ has coordinates zero, 10 and π΅ has coordinates five, zero.
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At this stage, it often helps to create a right-angled triangle on our graph.
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This will help us calculate the change in π¦ and the change in π₯, otherwise known as the rise and the run.
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Substituting our π¦-coordinates into the formula gives us zero minus 10.
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Substituting in our π₯-coordinates gives us five minus zero.
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It doesnβt matter which point is π₯ one, π¦ one and which is π₯ two, π¦ two.
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But we must be consistent in our order.
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Zero minus 10 is equal to negative 10.
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Five minus zero is equal to five.
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Negative 10 divided by five is equal to negative two.
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This means that the slope of the function represented on the graph is negative two.
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We can check this on the graph by considering the rise and the run.
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The rise is negative 10, as the π¦-coordinate drops from 10 to zero.
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The run is five, as the π₯-coordinate goes from zero to five.
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Once again, we have negative 10 divided by five.
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An important check is that any line that goes up from left to right will have a positive slope.
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Any line that goes down from left to right will have a negative slope.
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As our line goes downwards from left to right, a negative answer is sensible.
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We will now look at a second graph question.
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Calculate the slope of the line in the graph.
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We know that any straight line is a linear function that can be written in the form π¦ equals ππ₯ plus π, where π is the slope or gradient of the line.
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The value of π can be calculated using the formula π¦ two minus π¦ one over π₯ two minus π₯ one.
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This is the change in π¦-coordinates over the change in π₯-coordinates, otherwise known as the rise over the run.
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We begin by selecting any two points on the line π΄ and π΅ with coordinates π₯ one, π¦ one and π₯ two, π¦ two.
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Whilst it doesnβt matter which two points we choose, it is sensible to pick those with integer coordinates where possible.
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In this question, we will choose the two points shown on the graph.
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Point π΄ has coordinates zero, one and point π΅ has coordinates two, seven.
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At this point, it is worth drawing a right-angled triangle on the graph to show the rise and the run.
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The rise in this case is equal to six, as the change in π¦-coordinates is six.
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The run is equal to two.
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This means that we would expect the slope to be six divided by two, which is three.
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We can check this by substituting our coordinates into the formula.
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The two π¦-coordinates were seven and one.
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And the corresponding π₯-coordinates were two and zero.
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This simplifies to six over two, which once again gives us an answer of three.
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The slope of the line in the graph is three.
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It is worth recalling that any line that slopes upwards from left to right will have a positive slope.
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As three is positive, this suggests that our answer is correct.
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We will now look at a question that involves finding the slope of a linear function from a table.
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What is the slope of the linear function represented by the given table?
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We know that the equation of any linear function is written in the form π¦ equals ππ₯ plus π, where π is the slope or gradient of the function and π is the π¦-intercept.
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We can calculate the value of the slope π using the following formula, π¦ two minus π¦ one over π₯ two minus π₯ one.
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This is the change in π¦-coordinates over the change in π₯-coordinates, where any two points π΄ and π΅ have coordinates π₯ one, π¦ one and π₯ two, π¦ two, respectively.
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In our table, we have three coordinates, firstly, zero, four.
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Our second coordinate has an π₯-value of two and a π¦-value of 10.
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Our third coordinate, which we will call πΆ, is four, 16.
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We can select any two of these three coordinates.
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In this question, we will begin by considering point π΄ and point π΅.
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The π¦-coordinates of these two points are 10 and four.
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The corresponding π₯-coordinates are two and zero.
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The slope π is equal to 10 minus four over two minus zero.
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This simplifies to six over two, giving us a final answer of a slope of three.
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We can check this answer by selecting a different two points, in this case point π΄ and point πΆ.
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This time the slope is equal to 16 minus four over four minus zero.
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12 divided by four is also equal to three.
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We would get the same answer if we use the points π΅ and πΆ.
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The slope of the linear function represented by the table is three.
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We could also calculate this answer just by looking at the table.
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The change in π₯-values between the first and second point is plus two.
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The change in the π¦-values between the first two points is plus six.
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As the slope is equal to the change in π¦-values divided by the change in π₯-values, this also gives us an answer of three.
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For each single unit the π₯-value increases, the π¦-value will increase by three units.
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Our next question will include a graph in a real-world context.
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The graph shows the distance Amelia traveled over her two-hour bike ride.
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Which of the following is true?
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A) She traveled at a constant speed of four miles per hour for the last hour.
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B) She traveled at a constant speed of 10 miles per hour for the entire ride.
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C) She traveled at a constant speed of eight miles per hour for the last hour.
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Or D) she traveled at a constant speed of seven miles per hour for the entire ride.
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We can see from the graph that the π₯-axis represents the time in hours and the π¦-axis represents the distance in miles.
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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.
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If the graph is a straight line for the entire journey, then they will be traveling at a constant speed.
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We can see from the graph that three parts of the journey have different slopes or gradients.
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This means that, during these three parts, Amelia will be traveling at different speeds.
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We can therefore rule out options B and D, as these stated that she traveled at a constant speed for the entire ride.
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This is not the case as she will have traveled at three different speeds.
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Both of the other statements relate to the last hour of Ameliaβs journey.
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This occurs between the two points π΄ and π΅ on the graph.
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We can calculate the slope between any two points on a graph by using the following formula, π¦ two minus π¦ one over π₯ two minus π₯ one.
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This is the change in π¦-coordinates over the change in π₯-coordinates, in this case the change in the distance over the change in the time.
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Point π΄ has coordinates one, 10 and point π΅ has coordinates two, 14.
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The π¦-coordinates or distances here are 14 and 10.
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The corresponding π₯-coordinates are two and one.
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14 minus 10 is equal to four and two minus one is equal to one.
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This means that the slope of the line between points π΄ and π΅ is four.
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We could also have worked this out by drawing a right-angled triangle on the graph.
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We can see here that the distance has risen from 10 to 14.
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And the time has gone from one hour to two hours.
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Four divided by one is equal to four.
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So once again, the slope equals four.
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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.
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This rules out option C and therefore option A is correct.
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Amelia traveled at a constant speed of four miles per hour for the last hour.
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We will now recap some of the key points from this video.
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The graph of any linear function is a straight line.
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A linear function has a constant rate of change, which means that the difference in the π¦-coordinates of any two points on the straight line is proportional to the difference in their π₯-coordinates.
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This rate of change is the slope of the line.
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The equation of a line is generally written in the form π¦ equals ππ₯ plus π, where π is the slope or gradient of the line and π is the π¦-intercept.
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This is the point where the line crosses the π¦-axis.
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Finally, the slope of a line π is the rate of the vertical change to the horizontal change between two points.
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For two points π΄ π₯ one, π¦ one and π΅ π₯ two, π¦ two lying on a line, the slope is π, which is equal to π¦ two minus π¦ one over π₯ two minus π₯ one.
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If this number is positive, our line will slope upwards from left to right.
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And if it is negative, it will slope downwards from left to right.