# Transformation of Functions: Reflection

#### Vertical Reflection (Reflection about $x$-axis)

Given a function $f(x)$, a new function $g(x)=-f(x)$ is a horizontal reflection (reflection about the $x$-axis) of the function.

#### Horizontal Reflection (Reflection about $y$-axis)

Given a function $f(x)$, a new function $g(x)=-f(x)$ is a horizontal reflection (reflection about the $y$-axis) of the function.

#### Question (1)

The following are the sketch of the graphs of $y=f(x)$ and $y=g(x)$. Determine whether the graphs of $y=g(x)$ represents horizontal or vertical reflection of that of $y=f(x)$.

(a)

(b)

(c)

(d)

(e)

(f)

$\text{(a) }$ horizontal reflection

$\text{(b) }$ vertical reflection

$\text{(c) }$ vertical reflection

$\text{(d) }$ horizontal reflection

$\text{(e) }$ vertical reflection

$\text{(f) }$ horizontalreflection

#### Question (2)

Assume that $(a, b)$ is a point on the graph of $y=f(x)$. What is the corresponding point on the graph of each of the following functions?

$\begin{array}{l} \text{(a) } y=f(-x)\\\\ \text{(b) } y=-f(x)\\\\ \text{(c) } y=f(3-x)\\\\ \text{(d) } y=f(-x)-3 \end{array}$

$\begin{array}{l} \text{(a) }(-a, b)\\\\ \text{(b) } (a,-b)\\\\ \text{(c) } (3-a, b)\\\\ \text{(d) } (-a, b-3) \end{array}$

#### Question (3)

The figure shows the graph of $y=h(x)$.

Sketch the graphs of each of the following functions.

$\begin{array}{l} \text{(a) } y=-h(x)\\\\ \text{(b) } y=h(-x)\\\\ \text{(c) } y=h(-x)+2\\\\ \text{(d) } h(x-2) \end{array}$

(a) $y=-h(x)$

(b) $y=y=h(-x)$

(c) $y=y=h(-x)+2$

(d) $y=h(x-2)$

#### Question (4)

If $g$ is obtained from $f$ through a sequence of transformations, find an equation for $g$.

(a)

(b)

 $\text{(a)}$ The graph of $g$ is obtained by shifting the graph of $f$, 4 units left and then reflecting in $x$-axis. $g(x) = -f(x+4)$ $\text{(b)}$ The graph of $g$ is obtained by shifting the graph of $f$, 5 units left and 1 uint up and then reflecting in $x$-axis. $g(x) = -f(x-5)+1$

#### Question (5)

Which of the following is true?

 $\text{(a)}$ If $f(x)=|x|$ and $g(x)=|x+3|+3$, then the graph of $g$ is a translation of three units to the right and three units upward of the graph of $f$. $\text{(b)}$ If $f(x)=-\sqrt{x}$ and $g(x)=\sqrt{-x}$, then $f$ and $g$ have identical graphs. $\text{(c)}$ If $f(x)=x^{2}$ and $g(x)=-\left(x^{2}-2\right)$, then the graph of $g$ can be obtained from the graph of $f$ by a downward shift of two units and then reflecting in the $x$-axis. $\text{(d)}$ If $f(x)=x^{3}$ and $g(x)=-(x-3)^{3}-4$, then the graph of $g$ can be obtained from the graph of $f$ by moving $f$ three units to the right, reflecting in the $x$-axis, and then moving the resulting graph down four units.

$\begin{array}{l} \text{(a) false}\\\\ \text{(b) false}\\\\ \text{(c) true}\\\\ \text{(d) true} \end{array}$

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