Problem 32 - Algebra and Trigonometry

Question

\(27-32\) : A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=|x| ;\) shift to the left 1 unit, stretch vertically by a factor of \(3,\) and shift upward 10 units

Step-by-Step Solution

Answer

The final equation is \(k(x) = 3|x + 1| + 10\).

Step 1: Initial Function

We start with the given function, which is the absolute value function: \( f(x) = |x| \). This is a V-shaped graph centered at the origin.

Step 2: Horizontal Shift

The first transformation is a shift to the left by 1 unit. To shift a function to the left by \(1\) unit, we apply the transformation \(x' = x + 1\). So the function becomes \( g(x) = |x + 1| \).

Step 3: Vertical Stretch

Next, we apply a vertical stretch by a factor of \(3\). To vertically stretch a function, we multiply it by \(3\). Thus, the new function becomes \( h(x) = 3|x + 1| \).

Step 4: Vertical Shift

Finally, we shift the graph upward by \(10\) units. This involves adding \(10\) to the function, giving us \( k(x) = 3|x + 1| + 10 \).

Step 5: Final Transformed Function

After performing all the transformations, the final equation of the transformed graph is \( k(x) = 3|x + 1| + 10 \).

Key Concepts

Absolute Value Function

The absolute value function is a fundamental function in mathematics, denoted as \( f(x) = |x| \). It produces the non-negative value of \( x \), reflecting any negative input across the \( y \)-axis to make it positive. The graph of an absolute value function creates a V-shape, with its vertex typically located at the origin \((0,0)\). This characteristic V-shape makes the absolute value function easy to identify visually in graphs.

The basic properties of the absolute value function include:

• The domain is all real numbers \((-\infty, \infty)\).
• The range is all non-negative real numbers \([0, \infty)\).
• The function is symmetric about the y-axis, demonstrating even function properties.
• There is a sharp corner at the vertex, giving it the distinctive V shape.

This function is widely used in various fields, including physics, engineering, and economics, where the non-negativity of values is crucial.

Horizontal Shift

A horizontal shift involves moving the graph of a function left or right across the \( x \)-axis. For the function \( f(x) = |x| \), shifting it horizontally means changing the structure inside the absolute value brackets.

To shift the function \( f(x) = |x| \) to the left by 1 unit, we replace \( x \) with \( x + 1 \). This transformation results in a new function: \( g(x) = |x + 1| \). The notation \( x + 1 \) indicates that every point on the graph moves leftward by 1 unit.

Important points to understand about horizontal shifts:

• \( f(x + c) \) means shifting the function to the left by \( c \) units.
• \( f(x - c) \) moves the graph to the right by \( c \) units.

Horizontal shifts do not alter the shape of the graph; they only displace it laterally across the grid.

Vertical Stretch

A vertical stretch involves pulling the graph of a function away from the x-axis, effectively stretching it vertically. This is achieved by multiplying the function by a constant factor. For the transformed function \( g(x) = |x + 1| \) from the previous step, multiplying by 3 results in \( h(x) = 3|x + 1| \).

The transformation rule for vertical stretching is straightforward:

• Multiply the entire function by a positive constant \( a \). If \( a > 1 \), the graph stretches away from the x-axis.
• If \( 0 < a < 1 \), the graph compresses toward the x-axis, but this scenario deals specifically with stretching.

This vertical stretch increases the "height" of the V-shaped graph without altering its overall direction or symmetry. Each point's distance from the x-axis is multiplied by the stretching factor, making the graph appear taller.

Vertical Shift

In a vertical shift, the entire graph of a function moves up or down the y-axis. For our function \( h(x) = 3|x + 1| \), shifting it 10 units upward involves adding 10 to the entire function, resulting in \( k(x) = 3|x + 1| + 10 \).

This transformation has specific characteristics:

• Adding a positive constant shifts the graph upward.
• Subtracting a constant moves the graph downward.

Vertical shifts do not change the shape or horizontal position of the graph; they only elevate or lower every point by the same number of units.

In this final form, \( k(x) = 3|x + 1| + 10 \) reflects all the transformations combined: a leftward shift, a vertical stretch, and an upward shift, yielding the final position and shape of the graph.