Lesson 10

Combining Functions

  • Let’s make some new functions using other functions.

Problem 1

Sketch the graph of the function defined by the sum of function \(F\) and each of these functions.

Functions y of x and a of x.
Functions f of x and b of x.
Functions f of x and c of x.
Functions f of x and d of x.

Problem 2

Here are graphs of two functions \(f\) and \(g\). The function \(h\) is defined by \(h(x) = f(x) - g(x)\).

Graph of functions f and g.
  1. For which values of \(x\) is \(h(x) = 0\)? Explain how you know.
  2. For which values of \(x\) is \(h(x) < 0\)? For which values of \(x\) is \(h(x) > 0\)?
  3. Sketch a graph of \(h\).

Problem 3

The graph of each of these quadratic equations is a parabola. Explain how to find the vertex of each parabola by applying transformations to the graph of \(y = x^2\).

  1. \(y = (x-1)^2\)
  2. \(y = (x-1)^2 + 5\)
  3. \(y = \text-(x-1)^2 + 5\)
(From Unit 5, Lesson 7.)

Problem 4

The table shows the approximate United States population and the amount of sugar consumed in the given years.

year 1800 1850 1900 1950 2000
population (millions) 5 23 76 161 291
pounds of sugar consumed (millions) 30 460 3,000 12,000 32,000
  1. About how many pounds of sugar per year were consumed by each person in the 1800? 1850? 1900? 1950? 2000?
  2. Plot the numbers on the graph.

Blank coordinate plane, origin O. Horizontal axis from 0 to 200 by 50’s, labeled years after 1800. Vertical axis from 0 to 120 by 20’s, labeled pounds of sugar per person.

Problem 5

Here is a graph of a function \(f\).

  1. Use the graph to explain why this function is neither odd nor even.
  2. An equation defining \(f\) is \(f(x) = x^2 -2x\). Use the equation to verify that \(f\) is neither odd nor even.
Function on coordinate plane.
(From Unit 5, Lesson 6.)

Problem 6

In the table, we have the function \(f\) which gives Clare’s height \(f(t)\) above the ground, in feet, \(t\) seconds after starting her descent from the top of the original Ferris wheel. Today Clare tried out two new Ferris wheels. The first one, whose height is given by the function \(p\), is half the height of the original and turns at the same speed. The second one, whose height is given by the function \(r\), is the same height has the original but turns at twice the speed. 

\(t\) \(f(t)\) \(p(t)\) (half the height,
same speed)
\(r(t)\) (same height,
twice the speed)
0 212
5 181
10 106
15 31
20 0
  1. Fill in the table for functions \(p\) and \(r\).
  2. Explain why there is not enough information to determine \(r(15)\) and \(r(20)\).
  3. Express \(p\) and \(r\) in terms of \(f\).
(From Unit 5, Lesson 9.)