### Video Transcript

An object moves along a straight line. On the graph, the blue line shows the displacement đť‘‘ of the object from its starting position over time đť‘ˇ. Which of the three dashed lines is a tangent to the blue line at đť‘ˇ equals four seconds? (a) The red line, (b) the purple line, or (c) the orange line.

Letâ€™s first recall what a tangent is. A tangent is a straight line that touches a curve and has the same slope as the curve at the point where they touch. Now letâ€™s look at our graph, which has time đť‘ˇ along the horizontal axis and displacement đť‘‘ on the vertical axis. We are asked to find the tangent at đť‘ˇ equals four seconds. If we work upwards from the horizontal axis at đť‘ˇ equals four seconds, we find our blue line here with a displacement of eight meters. There is only one dashed line touching the curve at this point. And we can see that the purple dashed line has the same slope as the blue line. Therefore, the tangent here is (b) the purple line.

Next, weâ€™re asked to find the tangent to the blue line at đť‘ˇ equals one second. Again, we work up from đť‘ˇ equals one second on the horizontal axis to find the blue line. And we find there is only one dashed line touching the curve at this point. Here, the blue line has a much steeper slope than it had at đť‘ˇ equals four seconds. And we can see the slope of the blue line here is the same as the slope of the orange line. Therefore, at đť‘ˇ equals one second, the tangent is (c) the orange line.

Finally, which of the three dashed lines is a tangent to the blue line at đť‘ˇ equals 16 seconds? If we work upwards from the horizontal axis at đť‘ˇ equals 16 seconds, we find that the blue line has a much more shallow slope at this point. There is again just one dashed line touching the curve here. And you can see that it follows the slope of the blue line at this point. Therefore, the tangent here is (a) the red line.