KS4 Maths: Transformations of Trigonometric Graphs[Grade 8/9 - Free Printable

To match the graphs to their equations, we need to look at key features of each graph: the shape (sine, cosine, or tangent), the vertical shift (up or down), and any reflections (flips).

Here is the step-by-step breakdown for each equation:

1. $y = \sin(x)$
* Shape: Standard sine wave. Starts at $(0,0)$, goes up to $1$, down through $180^\circ$, to $-1$, and back to $0$ at $360^\circ$.
* Match: Graph A fits this perfectly.

2. $y = -\sin(x)$
* Shape: Reflected sine wave. Starts at $(0,0)$, but goes down first to $-1$, then up through $180^\circ$.
* Match: Graph E starts at $0$ and goes down.

3. $y = \sin(x) + 2$
* Shape: Sine wave shifted up by $2$. The middle line is $y=2$. It oscillates between $1$ and $3$.
* Match: Graph D oscillates between $y=1$ and $y=3$.

4. $y = \sin(x) - 3$
* Shape: Sine wave shifted down by $3$. The middle line is $y=-3$. It oscillates between $-2$ and $-4$.
* Match: Graph B oscillates between $y=-2$ and $y=-4$.

5. $y = \cos(x)$
* Shape: Standard cosine wave. Starts at the maximum value $(0,1)$, goes down to $-1$ at $180^\circ$, and back to $1$ at $360^\circ$.
* Match: Graph H starts at $y=1$ and follows the standard cosine pattern.

6. $y = -\cos(x)$
* Shape: Reflected cosine wave. Starts at the minimum value $(0,-1)$, goes up to $1$ at $180^\circ$, and back to $-1$ at $360^\circ$.
* Match: Graph C starts at $y=-1$.

7. $y = \cos(x) + 1$
* Shape: Cosine wave shifted up by $1$. Starts at $(0,2)$, goes down to $0$ at $180^\circ$, and back to $2$ at $360^\circ$.
* Match: Graph G starts at $y=2$ and touches the x-axis ($y=0$) at $180^\circ$.

8. $y = \cos(x) - 2$
* Shape: Cosine wave shifted down by $2$. Starts at $(0,-1)$, goes down to $-3$ at $180^\circ$, and back to $-1$ at $360^\circ$.
* Match: Graph F starts at $y=-1$ and goes down to $-3$.

9. $y = \tan(x)$
* Shape: Standard tangent curve. Passes through $(0,0)$, increases towards positive infinity as $x$ approaches $90^\circ$, and comes from negative infinity after $90^\circ$.
* Match: Graph K passes through the origin and has the correct increasing shape.

10. $y = -\tan(x)$
* Shape: Reflected tangent curve. Passes through $(0,0)$, but decreases towards negative infinity as $x$ approaches $90^\circ$.
* Match: Graph I passes through the origin but goes downwards.

11. $y = \tan(x) + 2$
* Shape: Tangent curve shifted up by $2$. The point that was at $(0,0)$ is now at $(0,2)$.
* Match: Graph J crosses the y-axis at $y=2$.

12. $y = \tan(x - 90)$
* Shape: This is a horizontal shift. $\tan(x - 90)$ is equivalent to $-\cot(x)$. At $x=0$, $\tan(-90)$ is undefined (vertical asymptote). As $x$ gets slightly larger than $0$, the value is large and negative. It crosses the x-axis at $x=90$.
* Match: Graph L has an asymptote at $x=0$ (going down on the right side) and crosses zero at $90^\circ$.

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Final Answer:
A: $y = \sin(x)$
B: $y = \sin(x) - 3$
C: $y = -\cos(x)$
D: $y = \sin(x) + 2$
E: $y = -\sin(x)$
F: $y = \cos(x) - 2$
G: $y = \cos(x) + 1$
H: $y = \cos(x)$
I: $y = -\tan(x)$
J: $y = \tan(x) + 2$
K: $y = \tan(x)$
L: $y = \tan(x - 90)$