Every Monday afternoon, Emily likes to go jogging in the park next to her job. She uses the pine tree seen in the diagram as her starting point to keep track of her progress.
Following her physics teacher's advice, Emily wrote and graphed a linear function d(t) that models the distance in meters to the right of the pine tree that she has jogged. Here, t is the time in seconds since the start of her run.
One day, she decided to invite Heichi to join her. Since Heichi is not used to jogging, Emily said that she will give him a head start. Emily will start 10 seconds later, and also she will start from the fountain, which she knows is 5 meters away from the pine.
Write the function rule g(t) that models Emily's jog, taking into account the advantage that she will give to Heichi.
Hint
Both a vertical and a horizontal translation will be needed.
Solution
Emily's jog is normally modeled by the following function.
d(t)=2.5t
Here, t is the time in seconds since Emily started jogging. Since she is giving Heichi a head start, she will start jogging 10 seconds after Heichi starts jogging. This means that at t=10, her distance jogged will be 0 meters, so the graph has to pass through (10,0). This can be done by translating the original graph 10 units to the right.
Furthermore, she will start from the fountain, 5 meters left of the pine tree, which is Heichi's starting point. Therefore, at t=10, Emily will be 5 meters to the left of the pine tree. This can be thought as her distance to the tree being -5 meters, so the graph has to pass through (10,-5). To achieve this, translate the previous graph 5 units down.
Therefore, the function g(t) that models Emily's new jog is obtained by translating d(t) 10 units to the right and 5 units down.
g(t)=d(t−10)−5⇓g(t)=2.5(t−10)−5
