Question

determine the amplitude and range of the function $y = -2\cos x$ and sketch its graph.
the amplitude is $\square$.

Explanation:

Step1: Identify amplitude formula

For $y = A\cos x$, amplitude is $|A|$.

Step2: Calculate the amplitude

Here $A=-2$, so amplitude = $|-2| = 2$.

Step3: Find the base range of $\cos x$

The range of $\cos x$ is $[-1, 1]$.

Step4: Scale range by $|A|$

Multiply all values by $2$: $[-2, 2]$. The negative sign reflects over the x-axis, which does not change the range.

Answer:

The amplitude is $\boldsymbol{2}$.
The range of the function is $\boldsymbol{[-2, 2]}$.

(For the graph: It is a cosine curve reflected over the x-axis, stretched vertically by a factor of 2, with a maximum value of 2, minimum value of -2, period of $2\pi$, passing through $(0, -2)$, $(\frac{\pi}{2}, 0)$, $(\pi, 2)$, $(\frac{3\pi}{2}, 0)$, $(2\pi, -2)$, and repeating every $2\pi$ units.)