Lesson 2

Moving Functions

  • Let’s represent vertical and horizontal translations using function notation.

2.1: What Happened to the Equation?

Graph each function using technology. Describe how to transform \(f(x)=x^2(x-2)\) to get to the functions shown here in terms of both the graph and the equation.

  1. \(h(x)=x^2(x-2)-5\)

  2. \(g(x)=(x-4)^2(x-6)\)

2.2: Writing Equations for Vertical Translations

The graph of function \(g\) is a vertical translation of the graph of polynomial \(f\).

Graph of a polynomial function \(f, x y\) plane. 
Graph of a polynomial function g, x y plane.
  1. Complete the \(g(x)\) column of the table.
  2. If \(f(0) = \text-0.86\), what is \(g(0)\)? Explain how you know.
  3. Write an equation for \(g(x)\) in terms of \(f(x)\) for any input \(x\).

  4. The function \(h\) can be written in terms of \(f\) as \(h(x)=f(x)-2.5\). Complete the \(h(x)\) column of the table.

    \(x\) \(f(x)\) \(g(x)\) \(h(x)=f(x)-2.5\)
    -4       0                       
    -3 -5.8    
    -0.7 0    
    1.2 -3.3    
    2 0    

  5. Sketch the graph of function \(h\).

    Blank coordinate x y plane with grid. Horizontal axis scale is negative 4 to 2, by 1’s. Vertical axis scale is negative 8 to 4, by 2’s. 
  6. Write an equation for \(g(x)\) in terms of \(h(x)\) for any input \(x\).

2.3: Heating the Kitchen

A bakery kitchen has a thermostat set to \(65^\circ \text{F}\). Starting at 5:00 a.m., the temperature in the kitchen rises to \(85^\circ \text{F}\) when the ovens and other kitchen equipment are turned on to bake the daily breads and pastries. The ovens are turned off at 10:00 a.m. when the baking finishes.

  1. Sketch a graph of the function \(H\) that gives the temperature in the kitchen \(H(x)\), in degrees Fahrenheit, \(x\) hours after midnight.
    Blank grid. Horizontal axis, hours after midnight, scale is 0 to 24 by 2’s. Vertical axis, degrees Fahrenheit, scale is 50 to 100 by 10’s.
  2. The bakery owner decides to change the shop hours to start and end 2 hours earlier. This means the daily baking schedule will also start and end two hours earlier. Sketch a graph of the new function \(G\), which gives the temperature in the kitchen as a function of time.
    Blank grid. Horizontal axis, hours after midnight, scale is 0 to 24 by 2’s. Vertical axis, degrees Fahrenheit, scale is 50 to 100 by 10’s.
  3. Explain what \(H(10.25) = 80\) means in this situation. Why is this reasonable?
  4. If \(H(10.25) = 80\), then what would the corresponding point on the graph of \(G\) be? Use function notation to describe the point on the graph of \(G\).
  5. Write an equation for \(G\) in terms of \(H\). Explain why your equation makes sense.


Write an equation that defines your piecewise function, \(H\), algebraically.

Summary

A pumpkin catapult is used to launch a pumpkin vertically into the air. The function \(h\) gives the height \(h(t)\), in feet, of this pumpkin above the ground \(t\) seconds after launch.

Now consider what happens if the pumpkin had been launched at the same time, but from a platform 30 feet above the ground. Let function \(g\) represent the height \(g(t)\), in feet, of this pumpkin. How would the graphs of \(h\) and \(g\) compare?

a photo of a wooden catapult on wheels 
Graph of three quadratic functions, time (seconds),height (feet).

Since the height of the second pumpkin is 30 feet greater than the first pumpkin at all times \(t\), the graph of function \(g\) is translated up 30 feet from the graph of function \(h\). For example, the point \((2,66)\) on the graph of \(h\) tells us that \(h(2) = 66\), so the original pumpkin was 66 feet high after 2 seconds. The new pumpkin would be 30 feet higher than that, so \(g(2) = 96\). Since all the outputs of \(g\) are 30 more than the corresponding outputs of \(h\), we can express \(g(t)\) in terms of \(h(t)\) using function notation as \( g(t) = h(t) + 30\).

Now suppose instead the pumpkin launched 5 seconds later. Let function \(k\) represent the height \(k(t)\), in feet of this pumpkin. The graph of \(k\) is translated right 5 seconds from the graph of \(h\). We can also say that the output values of \(k\) are the same as the output values of \(h\) 5 seconds earlier. For example, \(k(7) = 66\) and \(h(7-5) = h(2) = 66\). This means we can express \(k(t)\) in terms of \(h(t)\) as \(k(t)=h(t-5).\)