Problem 69 - College Algebra

Question

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ g(x)=4(x+1)^{2}-5 $$

Step-by-Step Solution

Answer

The function transforms \( x^2 \) by shifting left 1 unit, stretching by 4, and shifting down 5 units.

Step 1: Identify the Toolkit Function

The given function is \( g(x) = 4(x+1)^2 - 5 \). We begin by identifying the toolkit function, which is the base function before any transformations are applied. Here, the base function is \( f(x) = x^2 \), a standard quadratic function.

Step 2: Analyze Horizontal Shifts

The function \( g(x) = 4(x+1)^2 - 5 \) has \( (x+1) \) inside the parentheses. This indicates a horizontal shift. The \(+1\) inside the parentheses means that the graph is shifted 1 unit to the left from the toolkit function \( x^2 \).

Step 3: Analyze Vertical Stretches

The coefficient \( 4 \) in \( g(x) = 4(x+1)^2 - 5 \) indicates a vertical stretch. It stretches the graph of the base function \( x^2 \) vertically by a factor of 4.

Step 4: Analyze Vertical Shifts

The \(-5\) outside the quadratic term in \( g(x) = 4(x+1)^2 - 5 \) indicates a vertical shift. This moves the graph downward by 5 units.

Step 5: Sketch the Graph

Starting with the graph of \( f(x) = x^2 \), first shift it 1 unit to the left, then apply the vertical stretch by a factor of 4, and finally shift the graph downward by 5 units. These transformations create the graph of \( g(x) \).

Key Concepts

Horizontal Shift

In the function \( g(x) = 4(x+1)^2 - 5 \), we notice that inside the parentheses we have \((x + 1)\). This suggests a horizontal shift. A horizontal shift occurs when the input to a function changes, moving the graph left or right along the x-axis.

• A positive number added inside the parentheses, like \(+1\), shifts the graph to the left.
• Conversely, if there was a \(-\) sign, the graph would shift to the right.

For \(g(x)\), the graph of the toolkit function \(f(x) = x^2\) is shifted 1 unit to the left due to \((x + 1)\). This means every point of \(f(x)\) is repositioned one step leftward on the x-axis.

Vertical Stretch

A vertical stretch changes how tall or compressed a graph looks. It involves multiplying the function's output by a certain factor. In \(g(x) = 4(x+1)^2 - 5\), the presence of the coefficient \(4\) before \((x+1)^2\) indicates a vertical stretch.

• This coefficient amplifies the y-values by 4 times.
• The graph becomes four times steeper compared to the toolkit function \(f(x) = x^2\).

Think of a vertical stretch as stretching a rubber band vertically; it extends upward and downward, elongating the graph from its base form.

Vertical Shift

Vertical shifts are quite straightforward as they involve moving the graph up or down along the y-axis. In \(g(x) = 4(x+1)^2 - 5\), the \(-5\) indicates a vertical shift.

• A subtraction (like \(-5\)) shifts the graph downward by that many units.
• If it were a positive number, the graph would rise by that number of units.

In our case, the entire graph is moved 5 units down. This is added after all other transformations, ensuring the final shape is simply moved lower on the graph.

Toolkit Function

The foundation for understanding any transformed quadratic function is recognizing its toolkit function. The toolkit function is the simplest form of the function before any changes are made. For our example with \(g(x) = 4(x+1)^2 - 5\), the base or toolkit function is \(f(x) = x^2\).

• \(f(x) = x^2\) forms the basic parabola, a U-shaped graph that is symmetrical around the y-axis.
• Identifying the toolkit helps in understanding how subsequent transformations will shape the graph.

This foundational step allows students to see how the basic shape of the quadratics is adjusted through horizontal shifts, vertical stretches, and vertical shifts, integrating each aspect to form the final graph.