Lesson 7

Expressing Transformations of Functions Algebraically

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Problem 1

Here is a graph of \(f(x)=e^x\) and a graph of \(g\), which is a transformation of \(f\). Write an equation for the function \(g\).

2 functions on coordinate plane.

Solution

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Problem 2

Describe the transformation that takes the graph of function \(f\) to the graph of function \(g\).

  1. \(f(x)=e^x\) and \(g(x)=\text-e^x+2.7\)
  2. \(f(x)=x^5\) and \(g(x)=(\text-x+3.1)^5+1\)
  3. \(f(x)=|x|\) and \(g(x)=|x|-26\)
  4. \(f(x)=\sqrt x\) and \(g(x)=\text-\sqrt{x-0.004}\)

Solution

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Problem 3

  1. Write an equation whose graph is a parabola with vertex at \((1,4)\) and which opens upward.
  2. Write an equation whose graph is a parabola with vertex at \((2,\text-3)\) and which opens downward.

Solution

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Problem 4

Describe how to move the graph so that it better matches the data.

Set of data and line on x y grid.

Solution

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(From Unit 5, Lesson 1.)

Problem 5

Here is a graph of \(y = f(x)\) for \(\text-10 \le x \le 0\). Sketch \(f\) for \(0 \le x \le 10\) if:

  1. \(f\) is even
  2. \(f\) is odd
  3. \(f\) is neither even nor odd
Graph of function f.

Solution

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(From Unit 5, Lesson 6.)

Problem 6

Here are graphs of functions \(f\) and \(g\).

Which sequences of transformations take the graph of \(f\) to the graph of \(g\)? Select all that apply.

Functions g and f.
A:

reflection over the \(y\)-axis, then translation up by 2

B:

reflection over the \(x\)-axis, then translation up by 2

C:

translation up 2, then reflection over the \(y\)-axis

D:

translation up 2, then reflection over the \(x\)-axis

E:

translation up 2, and then translation left by 5

Solution

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(From Unit 5, Lesson 4.)