Porisma

Worked solution

Reflect output coordinates in the x-axis

Use a graphing calculator to sketch (1) y=(x+1)(x−3)y=(x+1)(x-3) and (2) y=−(x+1)(x−3)y=-(x+1)(x-3). Copy the curves. What do you observe about the curves?

CJC · Transformations · Reflection in x-axis activity

Sketch and compare the coordinates

Answer

The minimum (1,−4)(1,-4) becomes a maximum (1,4)(1,4); the roots stay (−1,0)(-1,0) and (3,0)(3,0).

The Teacher completes the Student observation table. Retain both curves on the same axes.

CurveStationary pointy-interceptx-intercepts
(x+1)(x−3)(x+1)(x-3)Minimum (1,−4)(1,-4)(0,−3)(0,-3)(−1,0),(3,0)(-1,0),(3,0)
−(x+1)(x−3)-(x+1)(x-3)Maximum (1,4)(1,4)(0,3)(0,3)(−1,0),(3,0)(-1,0),(3,0)
Expand diagram

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Overlaid original upward parabola with minimum (1,−4), y-intercept (0,−3), roots (−1,0),(3,0), and reflected downward parabola with maximum (1,4), y-intercept (0,3) and the same roots.
Overlaid original upward parabola with minimum (1,−4), y-intercept (0,−3), roots (−1,0),(3,0), and reflected downward parabola with maximum (1,4), y-intercept (0,3) and the same roots.
Overlaid original upward parabola with minimum (1,−4), y-intercept (0,−3), roots (−1,0),(3,0), and reflected downward parabola with maximum (1,4), y-intercept (0,3) and the same roots.
How did this part go?

Explain the reflection

Answer

y=−f(x)y=-f(x) reflects in the x-axis: (p,q)↦(p,−q)(p,q)\mapsto(p,-q).

Replace y by −y-y in y=f(x)y=f(x) to obtain −y=f(x)-y=f(x), hence y=−f(x)y=-f(x). Each output changes sign while the input stays fixed. As the source remark states, this is signed scaling by −1-1 parallel to the y-axis. Points on the x-axis stay fixed.

How did this part go?