Problem 69 - College Algebra with Corequisite Support

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 formula represents a parabola horizontally shifted left, vertically stretched by 4, and shifted 5 units downwards.

Step 1: Identify the Toolkit Function

The given function is based on the toolkit function of a parabola, which is the parent function: \( f(x) = x^2 \).

Step 2: Identify Horizontal Shifts

The expression \((x + 1)^2\) indicates a horizontal shift to the left by 1 unit. In general, \((x + h)^2\) translates the graph of \(x^2\) horizontally by \(-h\) units.

Step 3: Identify Vertical Stretches/Compressions and Reflections

The coefficient 4 in front of \((x + 1)^2\) indicates a vertical stretch. It means the parabola opens upwards and is steeper than \(x^2\), scaled by a factor of 4.

Step 4: Identify Vertical Shifts

The entire expression is subtracted by 5, which indicates a vertical shift downward by 5 units. This modifies the resulting graph of \(4(x + 1)^2\) by moving it 5 units down along the y-axis.

Step 5: Sketch the Graph

Start with the parent function \(f(x) = x^2\). Apply the horizontal shift 1 unit left, stretch the graph vertically by a factor of 4, and finally shift the entire graph downward by 5 units. The vertex of the parabola is at \((-1, -5)\) and it opens upwards.

Key Concepts

Toolkit Functions

To understand the transformation of the function \(g(x)=4(x+1)^{2}-5\), we start by identifying the base, or "toolkit" function. A toolkit function is a simple, well-known function that serves as a foundation for understanding transformations.

For quadratic functions, the toolkit function is the basic parabola, expressed as\(f(x) = x^2\).This function forms a U-shaped curve, symmetrical about the y-axis with its vertex at the origin (0,0). All transformations of quadratic functions, like shifting and stretching, are applied to this base function.

Horizontal Shifts

Horizontal shifts involve moving the graph left or right on the x-axis. This is determined by the expression inside the parentheses in our function.

In the term \((x + 1)^2\), the \(+1\) indicates a horizontal shift to the left. Generally, \(x + h\) in the function \((x + h)^2\) shifts the graph of \(x^2\) by \(-h\) units. This means for \(g(x) = 4(x+1)^2 - 5\), the entire graph moves left by 1 unit.Remember:

• If the expression is \(x - h\), the graph shifts right by \(h\) units.
• If it's \(x + h\), the graph shifts left by \(-h\) units.

Vertical Stretches

Vertical stretches or compressions modify how "tall" or "flat" a graph looks. This is influenced by the coefficient in front of the squared term. In \(g(x) = 4(x+1)^2 - 5\), the coefficient is \(4\).This applies a vertical stretch because the value is greater than 1, making the parabola steeper compared to \(x^2\).

The graph opens upwards and becomes taller, emphasizing the vertex at the shifted position. If the coefficient were less than 1, the graph would compress and be wider.

Vertical Shifts

Vertical shifts involve moving the entire graph up or down along the y-axis. These changes are indicated by constants added or subtracted from the function. In our equation\(g(x)=4(x+1)^{2}-5\), the \(-5\) directs a vertical shift downward by 5 units.

Here's a quick guide:

• Adding a constant will shift the graph upward by that value.
• Subtracting a constant shifts the graph downward.

Thus, the function's entire graph moves 5 units down, changing the vertex's y-coordinate in the process. The original vertex from \(x^2\) would have been at (0,0), however, after all transformations, it is now at (-1, -5).