A ball is thrown up in the air, and it falls back down to the ground. The height ℎ of the ball above the ground over time 𝑡 is shown on the graph by the blue line. Which of the five dashed lines shown on the graph is a tangent to the blue line at 𝑡 equals three seconds?
The graph is shown here on the bottom left with time in seconds on the horizontal axis and height in meters on the vertical axis. The path of the ball is shown by the blue line. So the height increases as it’s thrown up in the air and then decreases as the ball falls back to the ground. We are looking for a tangent to this blue line, so let’s recall the definition of a tangent.
A tangent is a straight line that touches a curve and has the same slope as the curve at the point where they touch. We’re looking for a tangent at 𝑡 equals three seconds. So let’s first find 𝑡 equals three seconds on our graph along the horizontal axis. We find it here. Working upwards from the horizontal axis, we find the point here on the blue line at 𝑡 equals three seconds, where we’re looking for a tangent. And we can see right away that all five of the dashed lines are straight lines and they all touch the curve at 𝑡 equals three seconds. So the key point is which one of these five dashed lines has the same slope as the blue line at 𝑡 equals three seconds.
Now we can see that at 𝑡 equals three seconds, the blue line is going downwards, meaning height is decreasing with time as the ball is falling back to the ground, which means we can immediately rule out the black line which is going upwards; i.e., height is increasing with time. Now, if we look at the green and orange lines, we can see these are both too shallow. Just before 𝑡 equals three seconds, they are beneath the blue line. And just after 𝑡 equals three seconds, they are above the line. Therefore, they cannot have the same slope.
The purple line has the opposite problem. Just before 𝑡 equals three seconds, it is above the blue line. And just after 𝑡 equals three seconds, it is below the blue line. Therefore, its slope is too steep. So now let’s look at the red line. The red line starts above the blue line, stays very close to it around 𝑡 equals three seconds, and then continues on above the blue line. Therefore, the line that has the same slope as the curve at the point 𝑡 equals three seconds is the red line. And so that is our tangent.