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Sketch the graph in the grid provided: \( f(x)=2 \cos ^{2} x-1 ; \quad x \in\left[-180^{\circ} ; 180^{\circ}\right] \)

Ask by Turnbull Weston. in South Africa
Feb 22,2025

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To sketch the graph of \( f(x) = 2 \cos^2 x - 1 \) for \( x \) between –180° and 180°: 1. **Identify the Function**: Recognize that \( f(x) = \cos 2x \). 2. **Key Features**: - **Amplitude**: 1 - **Period**: 180° - **Phase Shift**: 0 - **Vertical Shift**: 0 3. **Plot Key Points**: - \( x = -180° \): \( f(x) = 1 \) - \( x = -135° \): \( f(x) = 0 \) - \( x = -90° \): \( f(x) = -1 \) - \( x = -45° \): \( f(x) = 0 \) - \( x = 0° \): \( f(x) = 1 \) - \( x = 45° \): \( f(x) = 0 \) - \( x = 90° \): \( f(x) = -1 \) - \( x = 135° \): \( f(x) = 0 \) - \( x = 180° \): \( f(x) = 1 \) 4. **Draw the Graph**: - Plot these points on the grid. - Connect them with a smooth, continuous curve resembling the cosine function. - Ensure the graph is symmetric about the y-axis and repeats every 180°. This will give you the accurate sketch of the function within the specified interval.

Solución

We start with the given function:   f(x) = 2 cos²x – 1  for x ∈ [–180°, 180°]. Step 1. Recognize the Identity Notice that the double-angle formula for cosine tells us:   cos 2x = 2 cos²x – 1. Thus, f(x) = cos 2x. Step 2. Determine Key Features Since f(x) = cos 2x, we have: • Amplitude = 1 (since cos 2x varies from –1 to 1) • Period = 360° / 2 = 180° (because when x increases by 180°, 2x increases by 360°) • Phase Shift = 0 (no horizontal shift) • Vertical Shift = 0 Step 3. Find Important Points Let’s choose key x–values that divide one period into equal parts. We know that for a cosine function, the standard key points in one period are:   x   f(x) = cos 2x   --------------------------------------  –180°  cos(–360°) = 1  –135°  cos(–270°) = 0       (since cos 270° = 0)  –90°   cos(–180°) = –1  –45°   cos(–90°) = 0       (since cos 90° = 0)   0°   cos 0° = 1   45°   cos 90° = 0   90°   cos 180° = –1   135°  cos 270° = 0   180°  cos 360° = 1 Step 4. Sketching the Graph on the Grid 1. Draw a horizontal (x-axis) and vertical (y-axis) axis. 2. Mark the x-axis from –180° to 180°. 3. Label the y-axis with values –1, 0, and 1. 4. Plot the key points:   • (–180°, 1)   • (–135°, 0)   • (–90°, –1)   • (–45°, 0)   • (0°, 1)   • (45°, 0)   • (90°, –1)   • (135°, 0)   • (180°, 1) 5. Draw a smooth, continuous curve through these points. The curve will follow the shape of the cosine function, starting at a maximum at –180°, going down to a minimum at –90°, rising back to a maximum at 0°, and so on. Step 5. Observations • The graph is periodic with period 180°; hence, you will see two complete cycles over the interval from –180° to 180°. • The function is even (symmetric about the y-axis) because cosine is an even function. • It is helpful to draw a dotted line at y = 0 to guide your hand when sketching the curve. Using these steps, you can accurately sketch the graph of f(x) = 2 cos²x – 1 (or equivalently, cos 2x) on the provided grid.

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The Deep Dive

To sketch the graph of the function \( f(x) = 2 \cos^2 x - 1 \) in the range \( x \in [-180^{\circ}, 180^{\circ}] \), first recognize that this function relates to the cosine function. You can use the identity \( \cos^2 x = \frac{1 + \cos(2x)}{2} \), so \( f(x) \) can be transformed to \( f(x) = 2 \cdot \frac{1 + \cos(2x)}{2} - 1 = \cos(2x) \). This means you are actually sketching \( f(x) = \cos(2x) \). The cosine function has a period of \( 360^{\circ} \), but since we’re looking at \( \cos(2x) \), the period becomes \( 180^{\circ} \). Hence, the graph will repeat every \( 180^{\circ} \). The roots occur where \( \cos(2x) = 0 \), which is at \( x = -90^{\circ}, 0^{\circ}, 90^{\circ} \) within the given interval. At \( x = -90^{\circ} \) and \( x = 90^{\circ} \), \( f(x) \) equals \(-1\), and at \( x = 0^{\circ} \), \( f(x) \) equals \( 1\). The maximum value is \( 1 \) at every multiple of \( 90^{\circ} \) and the minimum value is \(-1\). Connect these points smoothly to represent the oscillating nature of the cosine function. Now that you have the key points and behavior of the function, you can sketch accordingly!

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