Lesson 14
Graphs That Represent Situations
- Let’s examine graphs that represent the paths of objects being launched in the air.
Problem 1
Here are graphs of functions \(f\) and \(g\).
Each represents the height of an object being launched into the air as a function of time.
- Which object was launched from a higher point?
- Which object reached a higher point?
- Which object was launched with the higher upward velocity?
- Which object landed last?
Problem 2
Technology required. The function \(h\) given by \(h(t) = (1-t)(8+16t)\) models the height of a ball in feet, \(t\) seconds after it was thrown.
- Find the zeros of the function. Show or explain your reasoning.
- What do the zeros tell us in this situation? Are both zeros meaningful?
- From what height is the ball thrown? Explain your reasoning.
- About when does the ball reach its highest point, and about how high does the ball go? Show or explain your reasoning.
Problem 3
The height in feet of a thrown football is modeled by the equation \(f(t) = 6 + 30t - 16t^2\), where time \(t\) is measured in seconds.
- What does the constant 6 mean in this situation?
- What does the \(30t\) mean in this situation?
- How do you think the squared term \(\text-16t^2\) affects the value of the function \(f\)? What does this term reveal about the situation?
Problem 4
The height in feet of an arrow is modeled by the equation \(h(t)=(1+2t)(18-8t)\), where \(t\) is seconds after the arrow is shot.
- When does the arrow hit the ground? Explain or show your reasoning.
- From what height is the arrow shot? Explain or show your reasoning.
Problem 5
Two objects are launched into the air.
- The height, in feet, of Object A is given by the equation \(f(t)=4+32t-16t^2\).
- The height, in feet, of the Object B is given by the equation \(g(t)=2.5+40t-16t^2\). In both functions, \(t\) is seconds after launch.
- Which object was launched from a greater height? Explain how you know.
- Which object was launched with a greater upward velocity? Explain how you know.
Problem 6
- Predict the \(x\)- and \(y\)-intercepts of the graph of the quadratic function defined by the expression \((x+6)(x-6)\). Explain how you made your predictions.
- Technology required. Check your predictions by graphing \(y=(x+6)(x-6)\).
Problem 7
Technology required. A student needs to get a loan of $12,000 for the first year of college. Bank A has an annual interest rate of 5.75%, Bank B has an annual interest rate of 7.81%, and Bank C has an annual rate of 4.45%.
- If we graph the amount owed for each loan as a function of years without payment, predict what the three graphs would look like. Describe or sketch your prediction.
- Use graphing technology to plot the graph of each loan balance.
- Based on your graph, how much would the student owe for each loan when they graduate from college in four years?
- Based on your graph, if no payments are made, how much would the student owe for each loan after 10 years?
Problem 8
Technology required. The functions \(f\) and \(g\) are given by \(f(x) = 13x + 6\) and \(g(x) = 0.1 \boldcdot (1.4)^x\).
- Which function eventually grows faster, \(f\) or \(g\)? Explain how you know.
- Use graphing technology to decide when the graphs of \(f\) and \(g\) meet.