Repeat a square-root and line motif

y=f(x)

Total marks: 4

Solution

(i)

Periodic graphs

A curve has equation $y=f(x)$ , where $f(x)=\sqrt{8x+1}$ for $0\le x\le3$ , $f(x)=11-2x$ for $3<x\le5$ , and $f(x)=f(x-5)$ for all real $x$ . Sketch the curve for $-2\le x\le8$ .

Explore this idea

Hint

Find the values at the ends of each piece: $x=0$ , $3$ , $5$ .
Sketch one period from $0$ to $5$ .
Copy it 5 units to the left and right to fill $[-2,8]$ .

Q5 periodic piecewise graphScroll across to see the full diagram.

Answer · (i)

(-2,5),(0,1),(3,5),(5,1),(8,5)

Explore the idea

One period, repeated

Switch on the features to see how the periodic graph is assembled.

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 = f(x)
Key points
A mid-arc check point

Ideas to try

Compare the shape on

[0,5]

with the shape on

[5,10]

(ii)

Transformations

On a separate diagram, sketch the curve with equation $y=f\left(\tfrac12x-1\right)$ for $-2\le x\le8$ .

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Hint

Write $\frac12x-1$ as a sequence of two steps on $x$ .
Translate first, then scale. Which direction does each go?
Track the key points: $(-2,5)$ , $(0,1)$ , $(3,5)$ , $(5,1)$ , $(8,5)$ .

Q5 transformed periodic graphScroll across to see the full diagram.

Answer · (ii)

y=3-x

Explore the idea

Slide right 1, then stretch by 2

Two steps turn f(x) into f(x/2 − 1). The window is the same in each, so you can see parts leave and enter.

Start: y = f(x)

The graph from part (i).

Marked points at this step, rounded to 6 significant figures.
Point	x	y
(0,1)	0	1
(3,5)	3	5
(5,1)	5	1

Graph at the selected step
Labelled key points

Ideas to try

The final period is 10 instead of 5. Where does the second period start?
Check that $(5,1)$ from step 1 has left the window by step 2.

Repeat a square-root and line motif

Solution

Periodic graphs

End values of one period

Shape of each piece

Repeat with period 5