For the curve y=x1/3, draw (i) y=-x1/3 (ii) y=x1/3+1 (iii) y=x1/3-1 (iv) y=(x+1)1/3

Exercise 1.4 · Q2

y=x^{1/3}

(i)

y=-x^{1/3}

(ii)

y=x^{1/3}+1

(iii)

y=x^{1/3}-1

(iv)

y=(x+1)^{1/3}

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Concept understanding — Graphing Functions using Transformations

Rather than plotting a complicated curve point by point, recognise it as a reflection, translation, or dilation of an already-known simpler curve. Reflection and translation keep the graph congruent (same size and shape); dilation changes the shape by stretching/compressing.

Reflection (mirror image about a line):

y=-f(x)\ \text{reflects } f \text{ about the } x\text{-axis};\qquad y=f(-x)\ \text{reflects } f \text{ about the } y\text{-axis};\qquad y=f^{-1}(x)\ \text{reflects } f \text{ about } y=x.

Translation (slide without changing shape), for $c,d>0$ :

y=f(x+c)\to\text{left by }c;\quad y=f(x-c)\to\text{right by }c;\quad y=f(x)+d\to\text{up by }d;\quad y=f(x)-d\to\text{down by }d.

Dilation (stretch/compress): multiplying $f$ by a constant $k>0$ scales the graph vertically -- $k>1$ stretches it away from the $x$ -axis, $0<k<1$ compresses it toward the $x$ -axis. Replacing $x$ by $kx$ inside $f$ instead rescales horizontally (e.g. $y=\sin2x$ compresses $y=\sin x$ horizontally by factor 2, halving the distance between successive $x$ -intercepts while keeping the same $\pm1$ amplitude).

Combining transformations, in order. To build $y=2\sin(x-1)+3$ from $y=\sin x$ : shift right by 1 (translation) $\to y=\sin(x-1)$ ; stretch vertically by 2 (dilation) $\to y=2\sin(x-1)$ ; shift up by 3 (translation) $\to y=2\sin(x-1)+3$ . The same layered approach handles any $y=k\!\cdot\! f(x\pm c)\pm d$ built from a known base curve $f$ (lines, $|x|$ , $x^2,\ x^3,\ x^{1/3},\ \sin x$ , etc.), always working from the innermost change (horizontal shift/reflection inside $f$ 's argument) outward (vertical stretch, then vertical shift).

Step 1 (i) $y=-x^{1/3}$ . Reflection of $y=x^{1/3}$ about the $x$ -axis (and since cube-root is also odd, $-x^{1/3}=(-x)^{1/3}$ ). Key points: $(1,1)\to(1,-1)$ , $(8,2)\to(8,-2)$ , $(0,0)$ fixed.

Step 2 (ii) $y=x^{1/3}+1$ . Shift UP by 1. Key points: $(0,0)\to(0,1)$ , $(1,1)\to(1,2)$ , $(-1,-1)\to(-1,0)$ .

Step 3 (iii) $y=x^{1/3}-1$ . Shift DOWN by 1. Key points: $(0,0)\to(0,-1)$ , $(1,1)\to(1,0)$ .

Step 4 (iv) $y=(x+1)^{1/3}$ . Shift LEFT by 1. Key points: $(0,0)\to(-1,0)$ , $(1,1)\to(0,1)$ , $(-1,-1)\to(-2,-1)$ .

✓Final answer

y=x^{1/3}

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Q.For the curve y=x1/3y=x^{1/3}y=x1/3, draw

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Q.For the curve $y=x^{1/3}$ , draw

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