Question 1165421
This problem involves understanding function transformations based on how the coordinates of points change.

Since the graph of $f(x)$ is not provided, I will define a simple, representative graph with clear points to illustrate the transformations.

## Example Graph: $y = f(x)$

Let's assume the graph of $f(x)$ has the following four distinct points, which define its shape :
* $P_1 = (-4, 4)$
* $P_2 = (0, -2)$
* $P_3 = (4, 2)$

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## (a) Plot the graph of $y = g(x)$

The transformation rule is: Every point $(a, b)$ on the graph of $y = f(x)$ becomes the point $(5a, b - 3/2)$ on the graph of $y = g(x)$.

Applying the transformation $(x, y) \to \left(5x, y - \frac{3}{2}\right)$ to our example points:

* $P_1(-4, 4) \to P'_1 \left(5(-4), 4 - \frac{3}{2}\right) = P'_1 (-20, 2.5)$
* $P_2(0, -2) \to P'_2 \left(5(0), -2 - \frac{3}{2}\right) = P'_2 (0, -3.5)$
* $P_3(4, 2) \to P'_3 \left(5(4), 2 - \frac{3}{2}\right) = P'_3 (20, 0.5)$

The graph of $y = g(x)$ connects the points $(-20, 2.5)$, $(0, -3.5)$, and $(20, 0.5)$.



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## (b) Express $g(x)$ in terms of $f(x)$

To find the function $y = g(x)$, we reverse the transformations applied to the coordinates.

Let $(X, Y)$ be a point on the graph of $g(x)$, and let $(x, y)$ be the corresponding point on the graph of $f(x)$.

The transformation rule is:
1.  $X = 5x$
2.  $Y = y - \frac{3}{2}$

We need to solve for $x$ and $y$ in terms of $X$ and $Y$:
1.  $x = \frac{1}{5} X$
2.  $y = Y + \frac{3}{2}$

Since $(x, y)$ is on the graph of $f(x)$, we have $y = f(x)$. Substitute the expressions for $x$ and $y$:
$$Y + \frac{3}{2} = f\left(\frac{1}{5} X\right)$$

Now, solve for $Y$:
$$Y = f\left(\frac{1}{5} X\right) - \frac{3}{2}$$

Replacing $Y$ with $g(X)$ and using $x$ as the standard variable:
$$\mathbf{g(x) = f\left(\frac{1}{5} x\right) - \frac{3}{2}}$$

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## (c) Describe the transformations

The expression $g(x) = f\left(\frac{1}{5} x\right) - \frac{3}{2}$ reveals two transformations applied to $f(x)$:

1.  **Horizontal Transformation (due to $\frac{1}{5}x$ inside $f$):**
    The $x$-coordinates are multiplied by 5, which corresponds to a **horizontal stretch by a factor of 5**.
    $$\left(x \to \frac{1}{5}x\right) \implies \text{Horizontal Stretch by } 5$$

2.  **Vertical Transformation (due to $-\frac{3}{2}$ outside $f$):**
    The constant $\frac{3}{2}$ is subtracted from the $y$-value, which corresponds to a **vertical shift downward by $\frac{3}{2}$ units (or $1.5$ units)**.
    $$\left(y \to y - \frac{3}{2}\right) \implies \text{Vertical Shift Down by } \frac{3}{2}$$