Problem 63 - College Algebra and Calculus: An Applied Approach

Question

The point \((3,9)\) on the graph of \(f(x)=x^{2}\) has been shifted to the point \((4,7)\) after a rigid transformation. Identify the shift and write the new function \(g\) in terms of \(f\).

Step-by-Step Solution

Answer

The graph of the function \(f(x) =x^2\) was shifted 1 unit to the right and 2 units down to create the new function. The new function \(g\) is \(g(x)= (x-1)^{2} - 2\).

Step 1: Identify the horizontal shift

The horizontal shift can be found by subtracting the original x-coordinate from the new x-coordinate. This gives: \(4 - 3 = 1\). Therefore, the horizontal shift is 1 unit to the right.

Step 2: Identify the vertical shift

The vertical shift can be calculated by subtracting the original y-coordinate from the new y-coordinate. This gives: \(7 - 9 = -2\). Therefore, our function has been moved down by 2 units.

Step 3: Write the new function g in terms of f

The new function \(g\) can now be written as \(g(x)=f(x - 1) - 2 = (x - 1)^{2} - 2\). The -1 in \(f(x-1)\) accounts for the 1 unit right shift, and the -2 at the end of the function accounts for the 2 unit downward shift.

Key Concepts

Horizontal Shift

A horizontal shift in a function occurs when the graph of the function moves left or right along the x-axis. It’s a transformation that adjusts the x-values of a function. When you subtract a number from the variable inside the function, it produces a shift to the right; for example, in the function \(g(x) = f(x - 1)\), the graph is shifted 1 unit to the right. Conversely, adding a number shifts the graph to the left.

Some important points to remember about horizontal shifts are:

• Subtracting a positive number inside the function (like \(f(x-a)\)) shifts the graph right by \(a\) units.
• Adding a positive number inside the function (like \(f(x+a)\)) shifts the graph left by \(a\) units.
• A horizontal shift doesn’t alter the shape of the graph, just its position.

In the example from the exercise, moving from \((3, 9)\) to \((4, 9)\) results in a change of x-value from 3 to 4. This indicates a shift of 1 unit to the right, reflected in the function \(g(x) = f(x-1)\). This right shift is due to subtracting 1 from each x-value in the function.

Vertical Shift

A vertical shift involves moving the graph of a function up or down along the y-axis. This happens by adding or subtracting a value after applying the function’s rule. A vertical shift changes the y-values of the graph but maintains the x-values unchanged.

Understanding vertical shifts can be simplified with these tips:

• Adding a positive number to a function (i.e., \(f(x) + b\)) results in a shift upwards by \(b\) units.
• Subtracting a positive number from a function (i.e., \(f(x) - b\)) shifts the graph downwards by \(b\) units.
• Vertical shifts do not affect the x-values or the shape of the graph.

In the exercise provided, the transformation results in the y-coordinate changing from 9 to 7. This signifies a downward shift of 2 units, represented in the transformed function as \(-2\) added at the end, forming \(g(x) = (x-1)^2 - 2\). The minus two here shows the graph has moved down 2 units.

Quadratic Functions

Quadratic functions are polynomial functions of the form \(f(x) = ax^2 + bx + c\). Their graphs are parabolas, which can open upwards or downwards, depending on the sign of the coefficient \(a\). In their simplest form, quadratic functions may look like \(f(x) = x^2\), where the vertex of the parabola is at the origin and it opens upwards.

Key features of quadratic functions include:

• The vertex, which is either the highest or lowest point, depending on how the parabola opens.
• The axis of symmetry, a vertical line that passes through the vertex, represented by the equation \(x = -\frac{b}{2a}\).
• The shape of the parabola remains consistent even when it undergoes transformations.

In our exercise, the original function \(f(x) = x^2\) underwent a transformation resulting in \(g(x) = (x-1)^2 - 2\). This new equation indicates a standard quadratic function shifted horizontally and vertically. Despite these shifts, the function remains a parabola, showcasing the constant shape of quadratic functions during rigid transformations.