An exponential reflected at its zero

y=|(1/3)^{2x}-18|

Total marks: 6

Solution

(i)

Sketching a modulus curve

Sketch the graph of $y=\left|\left(\tfrac13\right)^{2x}-18\right|$ , finding the exact intercepts.

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Hint

$\left(\frac13\right)^{2x}=3^{-2x}$ : an exponential decay.
Intercepts: $x=0$ for the $y$ -intercept; set the inside to 0 for the $x$ -intercept.
The inside tends to $-18$ as $x\to\infty$ , so the modulus gives the asymptote $y=18$ .

Absolute exponential graphScroll across to see the full diagram.

Answer · (i)

(0,17)

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Drop by 18, then fold

Subtract 18 from the decaying exponential, then take the absolute value.

Start: y = 3^(−2x)

The decaying exponential, crossing the $y$ -axis at 1.

Marked points at this step, rounded to 6 significant figures.
Point	x	y
(0, 1)	0	1

Graph at the selected step

Ideas to try

How does the

y

-intercept change between steps 2 and 3?

(ii)

Absolute-value inequalities

Without a calculator, solve $\left|\left(\tfrac13\right)^{2x}-18\right|\le9$ with full working, giving the exact interval or intervals in their simplest form.

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Hint

$|u|\le9$ means $-9\le u\le9$ .
Add 18 to isolate $3^{-2x}$ .
Write the bounds as powers of 3 and compare exponents.

Answer · (ii)

-3/2\le x\le-1

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Where is the folded curve below y = 9?

Switch on 'Solution' to shade the strip between the curve and y = 9, from x = −3/2 to x = −1.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

Y = |3^(−2x) − 18|
Y = 9
Intersections and the solution

Ideas to try

Why does the line y = 9 meet the folded curve exactly twice?

An exponential reflected at its zero

Solution

Sketching a modulus curve

$y$ -intercept

$x$ -intercept

Asymptote

Shape

Drop by 18, then fold

Absolute-value inequalities

Remove the modulus

Isolate the exponential

Write as powers of 3

Compare exponents