Reflections of Graphs
For $y=-f(x)$, we reflect $y=f(x)$ in the $x$-axis.
For $y=f(-x)$, we reflect $y=f(x)$ in the $y$-axis.
Example 1
Consider $f(x)=x^3-4x^2+4x$. On the same axes, sketch the graphs of $y=f(x)$ and $y=-f(x)$.
\( \begin{align} \displaystyle
f(x) &= x^3-4x^2+4x \\
&= x(x^2-4x + 4) \\
&= x(x-2)^2
\end{align} \)
Example 2
Consider $f(x)=x^3-4x^2+4x$. On the same axes, sketch the graphs of $y=f(x)$ and $y=f(-x)$.
\( \begin{align} \displaystyle
f(x) &= x^3-4x^2+4x \\
&= x(x^2-4x + 4) \\
&= x(x-2)^2
\end{align} \)
Example 3
The function $y=f(x)$ is transformed to $g(x)=-f(x)$. Find the point on $g(x)$ corresponding to the point $(3,0)$ on $f(x)$.
$-f(x)$ means that the graph is to reflect $y=f(x)$ in the $x$-axis, which the sign of $y$-value changes.
Thus the $y$-value $0$ was transformed to $0$.
Therefore the point $(3,0)$ was transformed from $(3,0)$.
Example 4
The function $y=f(x)$ is transformed to $g(x)=-f(x)$. Find the point on $f(x)$ that has been transformed to the point $(7,-1)$ on $g(x)$.
$-f(x)$ means that the graph is to reflect $y=f(x)$ in the $x$-axis, which the sign of $y$-value changes.
Thus the $y$-value $-1$ was transformed from $1$.
Therefore the point $(7,-1)$ was transformed from $(7,1)$.
Algebra Algebraic Fractions Arc Binomial Expansion Capacity Chain Rule Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume
Responses