Problem 88 - College Algebra with Modeling and Visualization

Question

The points \((-12,6),(0,8),\) and \((8,-4)\) lie on the graph of \(y=f(x)\). Determine three points that lie on the graph of \(y=g(x)\). \(g(x)=-2 f(x)\)

Step-by-Step Solution

Answer

The points are (-12, -12), (0, -16), and (8, 8).

Step 1: Understand the Transformation

The function \(g(x) = -2 f(x)\) implies that each point on the graph \(y = f(x)\) will be vertically scaled by a factor of \(-2\). This means each \(y\)-value will be multiplied by \(-2\).

Step 2: Transform Point (-12, 6)

Start with the point \((-12, 6)\). Apply the transformation by multiplying the \(y\)-value by \(-2\). The new \(y\)-value is: \((-2) \times 6 = -12\). The new point is \((-12, -12)\).

Step 3: Transform Point (0, 8)

Next, take the point \((0, 8)\). Apply the transformation by multiplying the \(y\)-value by \(-2\). The new \(y\)-value is: \((-2) \times 8 = -16\). The new point is \((0, -16)\).

Step 4: Transform Point (8, -4)

Finally, transform the point \((8, -4)\). Multiply the \(y\)-value by \(-2\) to apply the transformation. The new \(y\)-value is: \((-2) \times (-4) = 8\). The new point is \((8, 8)\).

Key Concepts

Vertical Scaling

Vertical scaling is a transformation that changes the "height" of a graph by multiplying all of the output values, or the y-values, by a constant factor. When we refer to the function transformation of the type \( g(x) = c \, f(x) \), the constant \( c \) determines how much the function \( f(x) \) is vertically stretched or compressed.

• If \( c \) is greater than 1, the graph stretches away from the x-axis.
• If \( c \) is between 0 and 1, it compresses towards the x-axis.
• If \( c \) is negative, it also reflects the graph over the x-axis, flipping it upside down.

In our specific case, since \( g(x) = -2 f(x) \), the graph of \( f(x) \) is both vertically stretched by a factor of 2 and reflected across the x-axis. To apply this scaling to a point \( (x, y) \), simply multiply the \( y \)-value by \(-2\). This gives a new set of coordinates that represent the transformed point.

Graph of a Function

The graph of a function is the visual representation of all the points \( (x, y) \) that satisfy the equation \( y = f(x) \). It is essentially a collection of these points in the xy-plane.When you begin with a function \( f(x) \), each x-value has a corresponding y-value, which forms a dot on this plane. Connecting these points smoothly as per the function's nature gives you its graph. Some key characteristics of a graph include:

• The domain: all possible x-values for which the function is defined.
• The range: all resulting y-values (outputs) when you plug in the domain into the function.
• Key points: such as intercepts, turning points, or any specific points of interest.

Transformations, like vertical scaling, do not alter the domain but can significantly affect the range and the appearance of the graph. They allow us to understand how the function's outputs are modified, helping us predict how the graph behaves without plotting every point.

Point Transformation

Point transformation involves changing specific coordinates to graphically modify or predict new positions of points on a transformed graph. It is particularly useful when you perform function transformations like vertical, horizontal shifts, scaling, or reflections.To transform a specific point \((x, y)\) on a graph of \( f(x) \) to a new function \( g(x) \), you typically adjust the coordinates based on the transformation rules. For vertical scaling:- You keep the x-coordinate unchanged.- You multiply the y-coordinate by the scaling factor.In the exercise, given the points \((-12, 6)\), \((0, 8)\), and \((8, -4)\), we transform these using the scaling factor \(-2\). The result is:

• \((-12, 6)\) becomes \((-12, -12)\).
• \((0, 8)\) becomes \((0, -16)\).
• \((8, -4)\) becomes \((8, 8)\).

By transforming each point, you develop the new graph for \( g(x) \), gaining insight into its overall shape and direction compared to the original function.