Question

2025-2026 mcr3u unit 2_ fun...
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mcr3u: unit 2: functions
lesson 1.6, 1.7, and 1.8 exploring transformations of parent functions
vertical transformations and reflections in the x-axis
$f(x) = af(k(x - d)) + c$
is a representation of all of the transformations that can affect any functions $f(x)$.
vertical transformations: transforming $f(x)$ to $af(x)$
consider $y = f(x)$ a parent function:
if $f(x) = x^2$, then \tif $f(x) = \sqrt{x}$ then \tif $f(x) = \frac{1}{x}$ then
for the transformation $y = af(x)$
if $f(x) = x^2$, then $y = $\tif $f(x) = \sqrt{x}$ then $y = $\tif $f(x) = \frac{1}{x}$ then $y = $
how can “a” transform the graph?

1. “a” can reflect the graph in the x axis $(a < 0)$
2. “a” can vertically compress the graph $(0 < a < 1)$
3. “a” can vertically expand the graph $(a > 1)$

mcr3u: unit 2: functions
reflections in the x-axis:
***note: $r$ in $x = $ reflection in the x-axis

Explanation:

To determine the transformed functions \( y = a f(x) \) for each parent function:

For \( f(x) = x^2 \):

Step 1: Apply the transformation \( y = a f(x) \)

Substitute \( f(x) = x^2 \) into \( y = a f(x) \).
\( y = a \cdot x^2 \)
So, \( y = a x^2 \)

For \( f(x) = \sqrt{x} \):

Step 1: Apply the transformation \( y = a f(x) \)

Substitute \( f(x) = \sqrt{x} \) into \( y = a f(x) \).
\( y = a \cdot \sqrt{x} \)
So, \( y = a \sqrt{x} \)

For \( f(x) = \frac{1}{x} \):

Step 1: Apply the transformation \( y = a f(x) \)

Substitute \( f(x) = \frac{1}{x} \) into \( y = a f(x) \).
\( y = a \cdot \frac{1}{x} \)
Simplify to get \( y = \frac{a}{x} \)

Answer:

s:

• If \( f(x) = x^2 \), then \( y = \boldsymbol{a x^2} \)
• If \( f(x) = \sqrt{x} \), then \( y = \boldsymbol{a \sqrt{x}} \)
• If \( f(x) = \frac{1}{x} \), then \( y = \boldsymbol{\frac{a}{x}} \)