Porisma

Worked solution

Scale output distances about the x-axis

Use a graphing calculator to sketch:

  1. y=(x+1)(x−3)y=(x+1)(x-3)
  2. y=12(x+1)(x−3)y=\frac12(x+1)(x-3)
  3. y=2(x+1)(x−3)y=2(x+1)(x-3)

Copy the curves. What do you observe about the curves?

CJC · Transformations · Vertical scaling activity

Sketch and retain the coordinate table

Answer

Vertices (1,−4)(1,-4), (1,−2)(1,-2), (1,−8)(1,-8); common roots −1,3-1,3.

Let f(x)=(x+1)(x−3)f(x)=(x+1)(x-3). Draw the original, f(x)/2f(x)/2 and 2f(x)2f(x) on the same axes. The Teacher completes:

CurveMinimumy-interceptx-intercepts
f(x)f(x)(1,−4)(1,-4)(0,−3)(0,-3)(−1,0),(3,0)(-1,0),(3,0)
f(x)/2f(x)/2(1,−2)(1,-2)(0,−3/2)(0,-3/2)(−1,0),(3,0)(-1,0),(3,0)
2f(x)2f(x)(1,−8)(1,-8)(0,−6)(0,-6)(−1,0),(3,0)(-1,0),(3,0)
Expand diagram

Use the scrollbars or arrow keys to inspect details.

Overlaid f,f/2,2f. Minima(1,−4),(1,−2),(1,−8); y-intercepts −3,−3/2,−6; common roots −1,3. Inputs fixed, outputs scaled.
Overlaid f,f/2,2f. Minima(1,−4),(1,−2),(1,−8); y-intercepts −3,−3/2,−6; common roots −1,3. Inputs fixed, outputs scaled.
Overlaid f,f/2,2f. Minima(1,−4),(1,−2),(1,−8); y-intercepts −3,−3/2,−6; common roots −1,3. Inputs fixed, outputs scaled.
How did this part go?

Explain the replacements and scale factors

Answer

f(x)/2f(x)/2 halves vertical distances; 2f(x)2f(x) doubles them.

The source replacement chains continue on PDF 8 / printed 80. Replacing y by 2y gives 2y=f(x)2y=f(x), hence y=f(x)/2y=f(x)/2, vertical factor 1/2. Replacing y by y/2 gives y/2=f(x)y/2=f(x), hence y=2f(x)y=2f(x), vertical factor 2. Input coordinates stay fixed; output coordinates are halved or doubled. “Shape preserved” means the curves remain parabolas, not that their vertical dimensions stay unchanged.

How did this part go?