Question

If \(f(x)=\frac{5}{8} \sqrt{-\frac{7}{12} x}\) and \(g(x)=-\frac{2}{5} \sqrt{6(x+3)}-4,\) what transformations could you apply to the graph of \(f(x)\) to create the graph of \(g(x) ?\)

Short answer

Left 3 units, horizontal scaling by \( \frac{1}{\sqrt{6}} \), vertical reflection and scaling by \( -2 \), shift down 4 units.

Step-by-step solution

Identify the Original Functions

Given functions are: 1. \( f(x) = \frac{5}{8} \sqrt{-\frac{7}{12} x} \) 2. \( g(x) = -\frac{2}{5} \sqrt{6(x+3)} - 4 \)

Analyze the Inner Transformations of g(x)

Inside the square root of \( g(x) \), change of \( x \) to \( x+3 \) indicates a horizontal shift to the left by 3 units. The factor \( 6 \) inside the square root shows horizontal scaling by a factor of \( \frac{1}{\sqrt{6}} \).

Compare the Coefficients Outside the Root

The coefficients outside the square root for \( f(x) \) and \( g(x) \) show that we have a change from \( \frac{5}{8} \) to \( -\frac{2}{5} \). This involves a vertical scaling and a reflection over the x-axis.

Analyze the Outer Transformations of g(x)

The term \( -4 \) in \( g(x) = -\frac{2}{5} \sqrt{6(x+3)} - 4 \) indicates a vertical shift downward by 4 units.

Combine All Transformations

Combining all the identified transformations: 1. Horizontal shift left by 3 units 2. Horizontal scaling by factor \( \frac{1}{\sqrt{6}} \) 3. Vertical reflection over the x-axis 4. Vertical scaling by factor \( \frac{2}{5} \frac{8}{5} \) (since \( \frac{5}{8} \) to \( -\frac{2}{5} \) is a scaling factor of \( \frac{5}{8} \div -\frac{2}{5} = -2/5 \) ) 5. Vertical shift down by 4 units.

Key concepts

Horizontal Shift

A horizontal shift involves moving the graph of a function left or right along the x-axis. For the given problem, we observe the transformation inside the square root of the function \( g(x) \) from \( x \) to \( x+3 \). This shift means that every point on the original graph \( f(x) \) is moved 3 units to the left.

Simply put, if we have a point \( (a, b) \) on \( f(x) \), then on \( g(x) \), this point becomes \( (a-3, b) \).

Think of it like shifting all of our data points along the x-axis by 3 units to the left.

Vertical Scaling

Vertical scaling changes the steepness or height of the graph by multiplying the function by a constant factor. In this problem, we see a change in coefficients from \( \frac{5}{8} \) to \( -\frac{2}{5} \). To understand vertical scaling, we can calculate the factor:

\[ \frac{5}{8} \div \left( -\frac{2}{5} \right) = -\frac{5}{8} \times \frac{5}{2} = -\frac{25}{16} \]

This means that each point on the graph \( f(x) \) is scaled vertically by \( -\frac{25}{16} \). Imagine stretching or compressing the graph upwards or downwards.

Reflection Over x-Axis

When a function is reflected over the x-axis, each point on the graph is mirrored over the x-axis. This is done by multiplying the entire function by -1.

In our given function \( g(x) \), you see the negative sign in \( -\frac{2}{5} \sqrt{6(x+3)} \), indicating this reflection.

Each point \( (a, b) \) from \( f(x) \) will transform to \( (a, -b) \) in \( g(x) \). This inversion flips the graph upside down along the x-axis.

Horizontal Scaling

Horizontal scaling adjusts the width of the graph by a constant factor. In \( g(x) \), there's a multiplication by \( 6 \) inside the square root, affecting the x-values. This indicates a horizontal scaling by a factor of \( \frac{1}{\sqrt{6}} \).

To visualize, each point \( (a, b) \) on \( f(x) \) changes to \( \left( \frac{a}{\sqrt{6}}, b \right) \).

This transforms the graph by compressing or stretching it horizontally.

Vertical Shift

Vertical shifts move the graph up or down along the y-axis. In \( g(x) \), the presence of \( -4 \) at the end indicates a vertical shift downward by 4 units.

Every point \( (a, b) \) on \( f(x) \) will change to \( (a, b-4) \).

Imagine translating the entire graph down by 4 units. This sort of transformation helps in adjusting the base level of the graph on the y-axis.