Download original notes
PAGE 1

4.4 Graphing Functions (part 2)

Symmetry: y-axis, origin

y-axis symmetry

When \( f(x) \) is even \( \rightarrow f(x) = f(-x) \)

Example: \( f(x) = x^2 \)

\[ f(-x) = (-x)^2 = x^2 = f(x) \]

graph:

Graph of y=x^2 showing symmetry across the y-axis with points at (-2, 4) and (2, 4).
Same thing on either side of y-axis

Another example: \( f(x) = \cos(x) \)

\[ f(-x) = \cos(-x) = \cos(x) = f(x) \]
Graph of y=cos(x) showing even symmetry. Peaks at x=0 and intercepts at x=pi/2 and -pi/2.
PAGE 2

origin symmetry

\( f(x) \) is odd \( \rightarrow f(-x) = -f(x) \)

example: \( f(x) = x^3 \)

\[ f(-x) = (-x)^3 = -x^3 = -f(x) \]
Graph of y=x^3 showing origin symmetry with points (-2, -8) and (2, 8).

whatever is in one quadrant is duplicated in the diagonally opposite quadrant

(point same distance away from origin on line \( y=x \))

Another example: \( f(x) = \sin(x) \)

\[ f(-x) = \sin(-x) = -\sin(x) = -f(x) \]
Graph of y=sin(x) showing odd symmetry. Passes through origin with peaks at pi/2 and -pi/2.
PAGE 3

Sketching Functions from the First Derivative

The first derivative alone is enough to get a reasonable sketch.

Example

Sketch \( f(x) \) if we know \( f'(x) = (x-2)(x+1)(x+5) \) only.

Use the given \( f'(x) \) to complete the inc/dec and rel. max/min step.

Find critical numbers:

\[ f'(x) = 0 \rightarrow (x-2)(x+1)(x+5) = 0 \]\[ x = -5, -1, +2 \]

\( f' \text{ DNE} \rightarrow \text{never} \)

Sign chart for f' showing negative, positive, negative, positive intervals between -5, -1, and 2.
  • dec: \( (-\infty, -5), (-1, 2) \)
  • inc: \( (-5, -1), (2, \infty) \)

rel min: at \( x = -5 \), at \( x = 2 \)

rel max: at \( x = -1 \)

\( y = ? \) we don't know since we don't have \( f(x) \)

We can make up \( y \)'s for these.

→ we will get the right shape but not the right points

answer is not unique

PAGE 4

One Possible Graph

rel. min at \( x = -5 \)

rel. min at \( x = 2 \)

rel. max at \( x = -1 \)

\( y = -10 \)

\( y = 3 \)

\( y = 5 \)

← made up numbers

Graph of f(x) with local mins at (-5, -10) and (2, 3), and local max at (-1, 5).

Equally Right Alternative

Alternative graph of f(x) with the same critical points but different vertical placement.
PAGE 5

Example: Analyzing Critical Numbers

Given the derivative function:
\[ f'(x) = \sin(3x) \] on the interval \( \left[ -\frac{4\pi}{3}, \frac{4\pi}{3} \right] \)

Interval Analysis:

\[ -\frac{4\pi}{3} \le x \le \frac{4\pi}{3} \]

\[ -4\pi \le 3x \le 4\pi \]

Convenient to know this since we are working with \( \sin(3x) \).

Find Critical Numbers

Set the derivative equal to zero:

\[ f'(x) = 0 \]

\[ \sin(3x) = 0 \]

\[ 3x = -4\pi, -3\pi, -2\pi, -\pi, 0, \pi, 2\pi, 3\pi, 4\pi \]

\[ x = -\frac{4\pi}{3}, -\pi, -\frac{2\pi}{3}, -\frac{\pi}{3}, 0, \frac{\pi}{3}, \frac{2\pi}{3}, \pi, \frac{4\pi}{3} \]

Sign Chart of \( f' \)

The following sign chart illustrates the behavior of the derivative and the resulting increasing/decreasing nature of the function.

Sign chart for f prime from -4pi/3 to 4pi/3 showing alternating positive and negative intervals.
PAGE 6

Relative Extrema

  • Relative maximums at: \( x = -\pi, -\frac{\pi}{3}, \frac{\pi}{3}, \pi \)
  • Relative minimums at: \( x = -\frac{2\pi}{3}, 0, \frac{2\pi}{3} \)

Graphical Representation

The graph below shows the function with its relative extrema labeled along the x-axis.

Coordinate graph of a periodic function with peaks and valleys labeled with their corresponding x-values.
PAGE 7

Example: Sketching with the First Derivative

Sketch \( f(x) = x^x \) using the first derivative.

Use logarithmic differentiation to find \( f'(x) \):

\[ y = x^x \]\[ \ln y = \ln x^x \]\[ \ln y = x \ln x \]

Now implicitly differentiate to find \( y' \):

\[ \frac{1}{y} y' = x \cdot \frac{1}{x} + \ln x \cdot 1 \]\[ = 1 + \ln x \]\[ y' = y(1 + \ln x) = x^x(1 + \ln x) \]

So, \( f'(x) = x^x(1 + \ln x) \)

\[ f' = 0 \rightarrow x^x = 0 \quad \text{or} \quad 1 + \ln x = 0 \]

\( x^x = 0 \)

never

\( 1 + \ln x = 0 \)

\( \ln x = -1 \)

\( x = e^{-1} \)

this is the only critical number

PAGE 8

Sign Analysis of \( f' \)

For \( x < 0 \), \( \ln x \) is not defined. We analyze the sign around the critical point \( x = e^{-1} \approx 0.37 \).

Sign chart for f' showing negative sign and downward arrow before x = e^-1, and positive sign and upward arrow after.

Relative Minimum

Relative minimum at \( x = e^{-1} \).

We know \( f(x) = x^x \), so we find the corresponding \( y \)-value:

\[ y = f(e^{-1}) = (e^{-1})^{e^{-1}} \approx 0.7 \]
Coordinate graph of f(x)=x^x with a labeled relative minimum at x=e^-1 and y=(e^-1)^(e^-1).