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Mathematics LibreTexts

1.3: Shifting and Reflecting

 

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1. Six Basic Functions

Below are six basic functions:

  1. SHIFT.HTM_txt_absx.gif
  2.  







  3. Memorize the shapes of these functions.

2. Horizontal Shifting

 Consider the graphs

\(y =\)

  • \((x+0)^2\)
  • \((x+1)^2\)
  • \((x+2)^2\)
  • \((x+3)^2\)

Exercise

Use the list features of a calculator to sketch the graph of 

\(y = \dfrac{1}{ [x - \{0,1,2,3\}] }\)

Horizontal Shifting Rules

  • Rule 1: \(f(x - a) = f(x)\) shifted \(a\) units to the right.
  • Rule 2: \(f(x + a) = f(x)\) shifted \(a\) units to the left.

3. Vertical Shifting

Consider the graphs

\(y =\)

  • \(x^3\)
  • \(x^3+ 1\)
  • \(x^3 + 2\)
  • \(x^3 + 3\)

Exercise

Use the list features of a calculator to sketch the graph of 

\(y = x^3 - \{0,1,2,3\}\)

Vertical Shifting Rules

  • Rule 3: \(f(x ) + a = f(x)\) shifted a units up.
  • Rule 4: \(f(x) - a = f(x)\) shifted a units down.

4. Reflecting About the x-axis

Consider the graphs of

 \(y = x^2\) and \(y = -x^2\).

SHIFT.HTM_txt_xReflect.gif

x-Axis Reflection Rule

Rule 5: \(-f(x) = f(x)\) reflected about the x-axis.

5. Reflecting About the y-axis

Exercise

Use the calculator to graph

\(y=\sqrt{x}\)

and
\(y=\sqrt{-x}\)

y-Axis Reflection Rule

Rule 6: \(f(-x ) = f(x)\) reflected about the y-axis.

6. Stretching and Compressing

Exercise

Graph the following:

\(y = \{1,2,3,4\}x^3\)

\(y = {1/2,1/3,1/4,1/5}x^3\)

Stretching and Compression rules:

  • Rule 7: \(cf(x ) = f(x)\) (for \(c > 1\)) stretched vertically.
  • Rule 8: \(cf(x ) = f(x)\) (for \(c < 1\)) compressed vertically.

Exercise

Graph the following

  1. \(y = x^2 - 10\)
  2. \(y = \sqrt{x - 2}\)
  3. \(y = -|x - 5| + 3\)

We will do some examples (including the graph of the winnings for the gambler and for the casino). 

7. Increasing and Decreasing Functions

Definition

A function is called increasing if as an object moves from left to right, it is moving upwards along the graph. Or equivalently,

If \(x < y\), then \(f(x) < f(y)\).

Example 1

The curve

\[y = x^2\] 

is increasing on \((0,\infty)\) and decreasing on \((-\infty,0)\).

Contributors