## Stretch Rule 1

For $y=pf(x)$, $p \gt 0$, the effect of $p$ is to vertically stretch the graph by a factor by $p$.

- If $p \gt 1$, it moves points of $y=f(x)$ further away from the $x$-axis.
- If $0 \lt p \lt 1$, it moves points of $y=f(x)$ closer to the $x$-axis.

## Stretch Rule 2

For $y=f(kx)$, $k \gt 0$, the effect of $k$ is to horizontally compress the graph by a factor of $k$.

- If $k \gt 1$, it moves points of $y=f(x)$ further away from the $x$-axis.
- If $0 \lt k \lt 1$, it moves points of $y=f(x)$ closer to the $x$-axis.

### Example 1

Given that the point $(-2,1)$ lies on $y=f(x)$, find the corresponding point on the image function $y=3f(2x)$.

$2x$ means that the graph is to horizontally compress by a factor of $2$.

Thus the $x$-value $-2$ transforms to $-1$.

$3f(2x)$ means that the graph is to vertically stretch by a factor of $3$.

Thus the $y$-value $1$ transforms to $3$.

Therefore the point $(-2,1)$ transforms $(-1,3)$.

### Example 2

Find the point which is moved to the point $(-7,3)$ under the transformation $y=3f(2x)$.

$2x$ means that the graph is to horizontally compress by a factor of $2$.

Thus the $x$-value $-7$ was transformed from $-14$.

$3f(2x)$ means that the graph is to vertically stretch by a factor of $3$.

Thus the $y$-value $3$ was transformed from $1$.

Therefore the point $(-7,3)$ was transformed from $(-14,1)$.

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorials Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume