Problem 68 - College Algebra

Question

Write a formula for the function g that results when the graph of a given toolkit function is transformed as described. The graph of \(f(x)=x^{2}\) is horizontally stretched by a factor of 3 , then shifted to the left 4 units and down 3 units.

Step-by-Step Solution

Answer

The formula for \(g(x)\) is \(g(x) = \frac{(x+4)^2}{9} - 3\).

Step 1: Identify the Base Function

The base function given is the quadratic function \(f(x) = x^2\). This is a common toolkit function known for its parabolic shape.

Step 2: Apply Horizontal Stretch

A horizontal stretch by a factor of 3 will affect the \(x\) coordinate in the input of the function. The transformation for a horizontal stretch involves dividing the \(x\) by the stretch factor. Therefore, the new function is \(g(x) = (\frac{1}{3}x)^2\).

Step 3: Apply Horizontal Shift

To shift a function left by 4 units, you add 4 to the \(x\) before applying the function. Therefore, substitute \(x\) with \(x+4\) into the function to get \(g(x) = (\frac{1}{3}(x+4))^2\).

Step 4: Apply Vertical Shift

To shift the function downward by 3 units, subtract 3 from the entire function. Thus the updated function becomes \(g(x) = (\frac{1}{3}(x+4))^2 - 3\).

Step 5: Simplify the Expression

Expand the expression to make it clearer. \(g(x) = \left(\frac{1}{3}(x+4)\right)^2 - 3 = \left(\frac{x+4}{3}\right)^2 - 3 = \frac{(x+4)^2}{9} - 3\).

Key Concepts

Quadratic Function

The quadratic function is a fundamental concept in algebra and is expressed in the simple form \( f(x) = x^2 \). Often referred to as a basic toolkit function, it has a distinctive parabolic shape.
This shape means that the graph of the function is a curve that opens upwards, making it symmetrical around the y-axis.
Key features of the quadratic function include the vertex, which is the point \( (0, 0) \) in this case, and the axis of symmetry, which is the vertical line passing through the vertex.

• Vertex: The point where the parabola changes direction.
• Axis of symmetry: A mirror line that cuts the parabola into two equal halves.
• Standard form: \( f(x) = ax^2 + bx + c \), but in our simple function, \( a = 1 \), and \( b \) and \( c \) are zero.

Quadratic functions form the basis for understanding more complex polynomial functions and their transformations.

Horizontal Stretch

The horizontal stretch is a transformation that alters the width of the graph horizontally.
When a function like \( f(x) = x^2 \) undergoes a horizontal stretch, the effect is seen on the input \( x \)-values.
This stretch manipulates the "speed" at which the function's value increases or decreases as it moves away from the y-axis.

• A horizontal stretch by a factor greater than 1 makes the graph wider.
• The transformation is achieved by multiplying \( x \) by the reciprocal of the stretch factor before applying the function.
• In this case, a factor of 3 is applied, making the new equation \( g(x) = \left(\frac{1}{3}x\right)^2 \).

This implies every point on the original curve now lies three times farther away from the y-axis, resulting in a gentler curve.

Vertical Shift

A vertical shift moves the graph of a function up or down without altering its shape.
It affects the y-coordinates of the function, thereby changing the height at which the curve is situated on the graph.

• To shift down by a certain number of units, subtract from the entire output \( f(x) \).
• This transformation is written as \( g(x) = f(x) - 3 \) when shifting down by 3 units.
• For our function, after including the horizontal stretch and shift, the equation becomes \( g(x) = \left(\frac{(x+4)}{3}\right)^2 - 3 \).

This vertical shift downward by 3 recalibrates the position of the curve while maintaining its parabolic nature.

Horizontal Shift

A horizontal shift moves a function's graph left or right along the x-axis.
It modifies the x-coordinates before applying the function, effectively sliding the entire graph in one horizontal direction.

• To shift left, add to the \( x \) inside the function: \( g(x) = f(x + 4) \).
• In our function, the graph is shifted left by 4 units, which means replacing \( x \) with \( x + 4 \).
• This results in the equation becoming \( g(x) = \left(\frac{1}{3}(x+4)\right)^2 \).

Such a transformation doesn't alter the graph's shape, just its position to the left on the x-axis, making the parabola appear to start earlier.