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Worked solution

Transform a cubic and find its exact intercepts

The curve y=x3y=x^3 is transformed by a translation of 2 units in the positive x-direction, followed by a stretch with scale factor 1/21/2 parallel to the y-axis, followed by a translation of 6 units in the negative y-direction. Find the equation of the new curve in the form y=f(x)y=f(x) and the exact coordinates of the points where this curve crosses the x- and y-axes. Sketch the new curve.

[N2010/1/5(i)]

CJC · Transformations · Example 11

Derive the new equation

Answer

y=12(x−2)3−6y=\frac12(x-2)^3-6.

The Teacher completes the intermediate equations and asks that the replacements be demonstrated when necessary. The printed word “stretch” uses factor 1/21/2; it halves vertical distances.

Translate right two

y=(x−2)3y=(x-2)^3

Replace x by x−2x-2.

Halve vertical distances

2y=(x−2)3⟹y=12(x−2)32y=(x-2)^3\quad\Longrightarrow\quad y=\frac12(x-2)^3

Replace y by 2y2y.

Translate down six

y+6=12(x−2)3⟹y=12(x−2)3−6y+6=\frac12(x-2)^3\quad\Longrightarrow\quad y=\frac12(x-2)^3-6

Replace y by y+6y+6.

How did this part go?

Find both exact intercepts

Answer

(0,−10)(0,-10) and (2+123,0)(2+\sqrt[3]{12},0).

For the y-intercept, set x=0x=0: y=12(0−2)3−6=−10y=\frac12(0-2)^3-6=-10. For the x-intercept, set y=0y=0: 0=12(x−2)3−60=\frac12(x-2)^3-6, so (x−2)3=12(x-2)^3=12 and x=2+123x=2+\sqrt[3]{12}. These coordinates are exact; a decimal approximation does not replace the cube root.

How did this part go?

Sketch the transformed cubic

Answer

An increasing cubic crossing (0,−10)(0,-10) and (2+123,0)(2+\sqrt[3]{12},0).

Retain the source curve and both labelled intercepts. Independently, its stationary inflection is (2,−6)(2,-6) and dy/dx=32(x−2)2≥0dy/dx=\frac32(x-2)^2\ge0; this explains the horizontal flattening and the increasing shape.

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Increasing cubic y=(x−2)³/2−6, with exact y-intercept (0,−10) and x-intercept (2+cube root of 12,0). It flattens at the independently identified stationary inflection (2,−6).
Increasing cubic y=(x−2)³/2−6, with exact y-intercept (0,−10) and x-intercept (2+cube root of 12,0). It flattens at the independently identified stationary inflection (2,−6).
Increasing cubic y=(x−2)³/2−6, with exact y-intercept (0,−10) and x-intercept (2+cube root of 12,0). It flattens at the independently identified stationary inflection (2,−6).
How did this part go?