Question 1165421: The graph of f(x) is shown below. https://latex.artofproblemsolving.com/d/b/a/dba24dce7c2734af95b18f61b73b2a32f213cd79.png
For each point (a,b) on the graph of y = f(x), the point ( 3a - 1, b/2) is plotted to form the graph of another function y = g(x). For example, (0,2) lies on the graph of y = f(x), so (3 * 0 - 1, 2/2) = (-1,1) lies on the graph of y = g(x).
(a) Plot the graph of y = g(x). Include the diagram in your solution.
(b) Express g(x) in terms of f(x).
(c) Describe the transformations that you would apply to the graph of y = f(x) to obtain the graph of y = g(x). For example, one transformation might be to stretch the graph horizontally by a factor of 5.
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! 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)**.
$$\left(y \to y - \frac{3}{2}\right) \implies \text{Vertical Shift Down by } \frac{3}{2}$$
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