The lines $y = x + 1$ and $y = -x - 7$ are the axes of symmetry of the function $f(x) = \frac{-2}{x + p} + q$. 4.1 Show that $p = 4$ and $q = -3$. 4.2 Calculate the $x$-intercept of $f$. 4.3 Sketch the graph of $f$. Clearly label ALL intercepts with the axes and the asymptotes.

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The lines $y = x + 1$ and $y = -x - 7$ are the axes of symmetry of the function $f(x) = \frac{-2}{x + p} + q$ - NSC Mathematics - Question 4 - 2021 - Paper 1

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The lines $y = x + 1$ and $y = -x - 7$ are the axes of symmetry of the function $f(x) = \frac{-2}{x + p} + q$. 4.1 Show that $p = 4$ and $q = -3$. 4.2 Calculate th... show full transcript

Worked Solution & Example Answer:The lines $y = x + 1$ and $y = -x - 7$ are the axes of symmetry of the function $f(x) = \frac{-2}{x + p} + q$ - NSC Mathematics - Question 4 - 2021 - Paper 1

Step 1

Show that $p = 4$ and $q = -3$.

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Answer

Given the lines of symmetry, we can equate their midpoints:

Set the equations equal to find the intersection:

$x + 1 = -x - 7$

Rearranging gives us:

$2x = -8 \implies x = -4$

Substituting this value into either equation to find $y$ :

$y = -4 + 1 = -3$

Thus, the point of symmetry is $(-4, -3)$ . This corresponds to:

$p + q = 1 \text{ (1)}$

Since the horizontal line of symmetry is $y = -3$ , we have:

$q = -3\text{ (2)}$

Substituting (2) into (1):

$p - 3 = 1 \implies p = 4$

Therefore, we have shown that $p = 4$ and $q = -3$ .

Step 2

Calculate the $x$-intercept of $f$.

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Answer

To find the $x$ -intercept, set $y = 0$ :

$0 = \frac{-2}{x + 4} - 3$

Solving, we get:

$\frac{-2}{x + 4} = 3\implies -2 = 3(x + 4)\implies -2 = 3x + 12$

Rearranging gives:

$3x = -14 \implies x = -\frac{14}{3}$

Hence, the $x$ -intercept is at $x = -\frac{14}{3}$ .

Step 3

Sketch the graph of $f$. Clearly label ALL intercepts with the axes and the asymptotes.

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Answer

To sketch the graph of the function:

Identify the asymptotes:
- Vertical asymptote occurs at $x = -4$ (where the denominator is zero).
- Horizontal asymptote as $x \to \pm \infty$ approaches $y = -3$ .
Intercepts:
- Y-intercept can be found by substituting $x = 0$ into $f(x)$ :
$f(0) = \frac{-2}{0 + 4} - 3 = -0.5 - 3 = -3.5$ $f (0) = \frac{- 2}{0 + 4} - 3 = - 0.5 - 3 = - 3.5$
- Therefore, the graph crosses the axes at $(0, -3.5)$ and $igg(-\frac{14}{3}, 0\bigg)$ .
Shape of the Graph: The graph will approach the asymptotes, approaching $(x, y) = (-4, -3)$ , with the correct asymptotic behavior.

Please include these points and lines when sketching the function.

The lines $y = x + 1$ and $y = -x - 7$ are the axes of symmetry of the function $f(x) = \frac{-2}{x + p} + q$ - NSC Mathematics - Question 4 - 2021 - Paper 1

Worked Solution & Example Answer:The lines $y = x + 1$ and $y = -x - 7$ are the axes of symmetry of the function $f(x) = \frac{-2}{x + p} + q$ - NSC Mathematics - Question 4 - 2021 - Paper 1

Show that $p = 4$ and $q = -3$.

Calculate the $x$-intercept of $f$.

Sketch the graph of $f$. Clearly label ALL intercepts with the axes and the asymptotes.

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