In the diagram, the graphs of $f(x) = ext{log}_a x$ and $g$ are drawn. Graph $g$ is the reflection of $f$ in the line $y = x$. Graph $f$ passes through the point $P(4; 2)$. $Q$ is the x-intercept of $f$ and $R$ is the y-intercept of $g$. 5.1 Write down the coordinates of $P'$, the image of $P$ on $g$. 5.2 Show that $a = 2$. 5.3 Write down the equation of $g$ in the form $y = ...$ 5.4 $T$ is a point on $f$ in the first quadrant where $TR$ is parallel to the x-axis. Calculate the area of $ riangle ART'$.

NSC Mathematics Functions: In the diagram, the graphs of $f

In the diagram, the graphs of $f(x) = ext{log}_a x$ and $g$ are drawn - NSC Mathematics - Question 5 - 2024 - Paper 1

Question 5

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In the diagram, the graphs of $f(x) = ext{log}_a x$ and $g$ are drawn. Graph $g$ is the reflection of $f$ in the line $y = x$. Graph $f$ passes through the point $P... show full transcript

Worked Solution & Example Answer:In the diagram, the graphs of $f(x) = ext{log}_a x$ and $g$ are drawn - NSC Mathematics - Question 5 - 2024 - Paper 1

Step 1

5.1 Write down the coordinates of $P'$, the image of $P$ on $g$.

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Answer

Since $g$ is the reflection of $f$ about the line $y = x$ , the coordinates of $P'(x; y)$ can be found by swapping the coordinates of $P(4; 2)$ . Therefore, the coordinates of $P'$ are $(2; 4)$ .

Step 2

5.2 Show that $a = 2$.

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Answer

We know that $f(4) = ext{log}_a 4 = 2$ . This can be rewritten as:

$ext{log}_a 4 = 2$

Rearranging gives:

$4 = a^2$

Solving for $a$ , we find:

$a = 2$ .

Step 3

5.3 Write down the equation of $g$ in the form $y = ...$

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Answer

Since $g$ is the reflection of $f(x) = ext{log}_2 x$ , its equation can be expressed as:

$y = 2^x$ .

Step 4

5.4 $T$ is a point on $f$ in the first quadrant where $TR$ is parallel to the x-axis. Calculate the area of $ riangle ART'$.

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Answer

To find the area of $riangle ART'$ , we can use the formula for the area of a triangle:

$ext{Area} = rac{1}{2} imes ext{base} imes ext{height}$

Where:

The base ( $RT$ ) is 2 units (length along the x-axis).
The height ( $PT'$ ) is 3 units (vertical distance from $P'$ to $R$ ).

Thus, the area is:

$ext{Area} = rac{1}{2} imes 2 imes 3 = 3 ext{ units}^2$ .

In the diagram, the graphs of $f(x) = ext{log}_a x$ and $g$ are drawn - NSC Mathematics - Question 5 - 2024 - Paper 1

Worked Solution & Example Answer:In the diagram, the graphs of $f(x) = ext{log}_a x$ and $g$ are drawn - NSC Mathematics - Question 5 - 2024 - Paper 1

5.1 Write down the coordinates of $P'$, the image of $P$ on $g$.

5.2 Show that $a = 2$.

5.3 Write down the equation of $g$ in the form $y = ...$

5.4 $T$ is a point on $f$ in the first quadrant where $TR$ is parallel to the x-axis. Calculate the area of $ riangle ART'$.

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