Problem 42 - Precalculus : Building Concepts and Connections

Question

Use the verbal description to find an algebraic expression for the function. The graph of the function \(g(t)\) is formed by vertically scaling the graph of \(f(t)=|t|\) by a factor of -2 and moving it to the left by 5 units.

Step-by-Step Solution

Answer

The function is \(g(t) = -2|t + 5|\).

Step 1: Vertical Scaling

To vertically scale a function, we multiply it by a factor. In this case, we are told to vertically scale by a factor of -2. So, we multiply the \(f(t)\) by -2. The result becomes \(g(t) = -2|t|\).

Step 2: Horizontal Shift

To move a function horizontally, we should modify its input. When it is said to 'move left' by 5 units, it means that we need to replace \(t\) with \(t+5\) in our function. Therefore, \(g(t) = -2|t+5|\). This function shifts the function to the left by 5 units.

Step 3: Present the Final Function

After applying both transformations, vertical scaling and horizontal shift, we have the new function. So, the function \(g(t)\) that represents the transformations of \(f(t) = |t|\) is \(g(t) = -2|t + 5|\).

Key Concepts

Vertical Scaling

Vertical scaling is a process that stretches or compresses a graph up or down. It changes the height of the graph but not the width. When an absolute value function like \(f(t) = |t|\) undergoes vertical scaling, each value of the function is multiplied by a constant factor. This can either stretch the graph if the factor is greater than 1 or compress it if the factor is between 0 and 1.

In our exercise, the graph is scaled by a factor of -2. This will flip the graph across the x-axis and stretch it by a factor of 2. You change all the y-values of the original function by multiplying them by -2. So, if the original graph moves upwards, now it will move downwards twice as fast. The new scaled function becomes \(g(t) = -2|t|\).

In simpler terms:

• "-" indicates a reflection across the x-axis.
• "2" indicates a doubling in the stretch vertically.

Absolute Value Function

An absolute value function describes a graph that has a characteristic 'V' shape. The basic form of this function is \(f(t) = |t|\). The absolute value symbol ensures all outputs are non-negative, as it reflects any negative inputs to positive values.

This function plays a crucial role because its transformations are easy to visualize. For any given input 't', the output is the distance of 't' from zero on the real number line, always positive. The vertex of the basic absolute value function is at the origin, i.e., (0,0).

When you transform an absolute value function using vertical scaling or horizontal shifts, these transformations modify the vertex's position and the graph's shape without altering the fundamental 'V' pattern.

Key Points to Remember:

• Always outputs positive values.
• The graph is symmetric around the y-axis for the basic form \(|t|\).

Horizontal Shift

A horizontal shift modifies the position of a graph left or right on the coordinate plane. To apply this shift, you adjust the input variable of the function. When you move a graph to the left, you replace each \(t\) with \(t + \text{constant}\). For a rightward shift, you replace \(t\) with \(t - \text{constant}\).

In our unique transformation, the graph is moved left by 5 units by substituting \(t\) with \(t + 5\). As a result, this change directly affects the x-position of the absolute value function’s vertex, shifting it from \(0\) to \(-5\). The newly adjusted function becomes \(g(t) = -2|t + 5|\).

Quick Recap:

• Left shift: replace \(t\) with \(t + \text{constant}\).
• Right shift: replace \(t\) with \(t - \text{constant}\).
• Horizontal shifts alter the graph’s starting point without changing its shape.