04Aexpgraphs

toc = Exponential Graphs =


 * Exponential graphs ** are formed by graphing an exponential function.

An ** exponential function ** has the variable in the exponent (or index or power).

Basic Exponential Graphs
The graph of y = a x (where a is positive but excluding 1) is the basic ** exponential graph **.

media type="custom" key="7978864"media type="custom" key="7978872" align="right" The graph of y = 2 x is shown media type="custom" key="7978932":

The graph of y = 3 x is shown media type="custom" key="7978886":

The graph of y = 4 x is shown media type="custom" key="7978894":

All of these exponential graphs have:

Asympote: y = 0

y-intercept: y = 1

No x-intercepts

No turning points

2nd point at (1, a)

Domain: x Î R

Range: y Î R+

These are strictly increasing graphs (gradient is always positive). include component="page" page="zHTML_div_close"

Exponential Graphs : a > 1
All exponential graphs, y = a x, with a > 1 have the following in common:
 * Asymptote at y = 0: as x → – ¥, y → 0
 * y-intercept at (0, 1): a 0 = 1
 * 2nd point at (1, a): a 1 = a
 * Strictly Increasing graphs
 * As x → + ¥, y → + ¥
 * Domain: x Î R
 * Range: y Î R+.



Exponential Graphs : 0 < a < 1
An exponential graph with fractional a (between 0 and 1) has the same shape but is reflected across the y-axis.

All exponential graphs, y = a x, with 0 < a < 1 have the following in common:
 * Asymptote at y = 0: as x → + ¥, y → 0
 * y-intercept at (0, 1): a 0 = 1
 * 2nd point at (1, a): a 1 = a
 * **Strictly Decreasing** graphs:
 * As x → – ¥, y → + ¥
 * Domain: x Î R
 * Range: y Î R+.

math \text{Note that: } 0.5^x = \left( \frac{1}{2} \right)^x = 2^{-x} math

so using the transformation rules, we can see that:

y = 0.5 x is the reflection of y = 2 x across the y-axis (in the x-direction)

math \text{Similarly } 0.1^x = \left( \frac{1}{10} \right)^x = 10^{-x} \text{ etc} math

Transformations
We can apply the standard transformations such as dilations, translations and reflections to any exponential graph, y = a x.

** Example 1 **
Sketch y = 3 × 2 x + 1 – 2


 * Method 1 **

Horizontal asymptote when 2 a = 0 y = 3 × 0 – 2 y = –2

y-intercept at x = 0 y = 3 × 2 1 – 2

y = 4 **®** (0, 4)

{At this point we'd usually look at 2 1 but we've already done that}

A 2nd point at 2 0 = 1 x + 1 = 0 x = –1 y = 3 × 1 – 2 y = 1 **®** (–1, 1) x-intercept when y = 0 math \\ 3 \times 2^{x+1} - 2 = 0 \\ \\ 3 \times 2^{x+1} = 2 \\ \\ 2^{x+1} = \frac{2}{3} math

{Convert to log form} math \\ x+1 = \log_2 \left( \frac{2}{3} \right) \\ \\ x = \log_2 \left( \frac{2}{3} \right) - 1 math

{This is an acceptable answer, but we could simplify further} math \\ x = \log_2(2) - \log_2(3) - 1 \\ \\ x = 1 - \log_2(3) - 1 \\ \\ x = -\log_2(3) math


 * Method 2 **

y = 3 × 2 x + 1 – 2

This is the graph of y = 2 x with transformations:
 * Dilation by a factor of 3 from the x-axis (in the y-direction) {3 × y}
 * Translation by 1 unit to the left {x – 1}
 * Translation by 2 units down {y – 2}

In matrix form, this is: math \left[ \begin{matrix} x' \\ y' \\ \end{matrix} \right] = \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \\ \end{matrix} \right] \, \left[ \begin{matrix} x \\ y \\ \end{matrix} \right] + \left[ \begin{matrix} -1 \\ -2 \\ \end{matrix} \right] math or math \\ x' = x - 1 \\ \\ y' = 3y - 2 math

Asymptote: y = 0 **®** y' = –2

Original y-intercept: (x = 0, y = 1) **®** (x' = –1, y' = 1)

Original 2nd point: (x = 1, y = 2) **®** (x' = 0, y' = 4)

Locate x-intercept as shown above. Go to top of page flat

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