Given $$f(x) = an \left( \frac{1}{2} x \right)$$ and $$g(x) = \sin \left( x - 30^{\circ} \right) \text{ for } x \in [-90^{\circ}; 180^{\circ}]$$ 6.1 On the same set of axes draw the graphs of $f$ and $g$. Show clearly on your graphs the turning points and asymptotes, if any. 6.2 Write down the period of $f$. 6.3 For what values of $x$ is $f(x).g(x) < 0$ for $x \in [-90^{\circ}; 120^{\circ}]$? 6.4 Write down the equation(s) of the asymptotes of $h$ if $h(x) = f(x + 10^{\circ})$ for $x \in [-90^{\circ}; 180^{\circ}]$.

NSC Mathematics Trigonometry: Given $$f(x) = an \left( \fra

Given $$f(x) = an \left( \frac{1}{2} x \right)$$ and $$g(x) = \sin \left( x - 30^{\circ} \right) \text{ for } x \in [-90^{\circ}; 180^{\circ}]$$ 6.1 On the same set of axes draw the graphs of $f$ and $g$ - NSC Mathematics - Question 6 - 2017 - Paper 2

Question 6

$Given---$$f(x)-=--an-\left(-\frac{1}{2}-x-\right)$$--and---$$g(x)-=-\sin-\left(-x---30^{\circ}-\right)-\text{-for-}-x-\in-[-90^{\circ};-180^{\circ}]$$--6.1-On-the-same-set-of-axes-draw-the-graphs-of-$f$-and-$g$-NSC Mathematics-Question 6-2017-Paper 2.png$

Given $$f(x) = an \left( \frac{1}{2} x \right)$$ and $$g(x) = \sin \left( x - 30^{\circ} \right) \text{ for } x \in [-90^{\circ}; 180^{\circ}]$$ 6.1 On the sa... show full transcript

Worked Solution & Example Answer:Given $$f(x) = an \left( \frac{1}{2} x \right)$$ and $$g(x) = \sin \left( x - 30^{\circ} \right) \text{ for } x \in [-90^{\circ}; 180^{\circ}]$$ 6.1 On the same set of axes draw the graphs of $f$ and $g$ - NSC Mathematics - Question 6 - 2017 - Paper 2

Step 1

6.1 On the same set of axes draw the graphs of f and g.

96%

114 rated

Answer

To draw the graphs of the functions, we start with the graph of $f(x) = \tan \left( \frac{1}{2} x \right)$ . The function has vertical asymptotes at the points where the tangent function is undefined, which occur at:

$x = (2n + 1) \cdot 90^{\circ}$

for integer values of $n$ . For our interval, the asymptotes occur at $x = -90^{\circ}$ and $x = 90^{\circ}$ . The graph will oscillate between these points, with turning points occurring within these intervals.

For the function $g(x) = \sin(x - 30^{\circ})$ , the graph will oscillate between -1 and 1, with a phase shift of $30^{\circ}$ . We can plot this on the same axes, showing how both functions intersect, their turning points, and the asymptotes for $f(x)$ .

Step 2

6.2 Write down the period of f.

99%

104 rated

Answer

The period of the function $f(x) = \tan \left( \frac{1}{2} x \right)$ can be calculated as:

$\text{Period}(f) = \frac{180^{\circ}}{\frac{1}{2}} = 360^{\circ}.$

Step 3

6.3 For what values of x is f(x).g(x) < 0 for x \in [-90^{\circ}; 120^{\circ}]?

96%

101 rated

Answer

To find the values of $x$ for which $f(x) \cdot g(x) < 0$ , we analyze the signs of both functions over the interval $[-90^{\circ}; 120^{\circ}]$ .

For $x$ in the interval $[-90^{\circ}, 0^{\circ}]$ , $f(x)$ is negative and $g(x)$ is negative.
For $x$ in the interval $(0^{\circ}, 30^{\circ})$ , $f(x)$ is positive and $g(x)$ is negative.

Thus, the solution for when $f(x) \cdot g(x) < 0$ occurs in:

$x \in (0^{\circ}, 30^{\circ}).$

Step 4

6.4 Write down the equation(s) of the asymptotes of h if h(x) = f(x + 10^{\circ}) for x \in [-90^{\circ}; 180^{\circ}].

98%

120 rated

Answer

The function $h(x) = f(x + 10^{\circ}) = \tan \left( \frac{1}{2} (x + 10^{\circ}) \right)$ will have asymptotes where:

$\frac{1}{2} (x + 10^{\circ}) = (2n + 1) \cdot 90^{\circ}$

Solving for $x$ gives:

$x + 10^{\circ} = (2n + 1) \cdot 180^{\circ}$

$x = (2n + 1) \cdot 180^{\circ} - 10^{\circ}.$

The specific asymptotes in the interval $[-90^{\circ}; 180^{\circ}]$ will be determined by selecting integer values for $n$ .

Given $$f(x) = an \left( \frac{1}{2} x \right)$$ and $$g(x) = \sin \left( x - 30^{\circ} \right) \text{ for } x \in [-90^{\circ}; 180^{\circ}]$$ 6.1 On the same set of axes draw the graphs of $f$ and $g$ - NSC Mathematics - Question 6 - 2017 - Paper 2

Worked Solution & Example Answer:Given $$f(x) = an \left( \frac{1}{2} x \right)$$ and $$g(x) = \sin \left( x - 30^{\circ} \right) \text{ for } x \in [-90^{\circ}; 180^{\circ}]$$ 6.1 On the same set of axes draw the graphs of $f$ and $g$ - NSC Mathematics - Question 6 - 2017 - Paper 2

6.1 On the same set of axes draw the graphs of f and g.

6.2 Write down the period of f.

6.3 For what values of x is f(x).g(x) < 0 for x \in [-90^{\circ}; 120^{\circ}]?

6.4 Write down the equation(s) of the asymptotes of h if h(x) = f(x + 10^{\circ}) for x \in [-90^{\circ}; 180^{\circ}].

Join the NSC students using SimpleStudy...