Problem 9 - College Algebra with Modeling and Visualization

Question

Find an equation that shifts the graph of \(f\) by the desired amounts. Do not simplify. Graph \(f\) and the shifted graph in the same \(xy\)-plane. \(f(x)=x^{2} ;\) right 2 units, downward 3 units

Step-by-Step Solution

Answer

The shifted equation is \(g(x) = (x - 2)^2 - 3\).

Step 1: Original Function

The given function is \(f(x) = x^2\). This represents a parabola with its vertex at the origin (0, 0) and opens upwards.

Step 2: Shift Right by 2 Units

To shift a function horizontally, you replace \(x\) with \(x - h\). For a shift to the right by 2 units, replace \(x\) with \(x - 2\). Thus, the function becomes \(f(x) = (x - 2)^2\).

Step 3: Shift Downward by 3 Units

To shift a function vertically downward, you subtract \(k\) from the function. For a shift downward by 3 units, subtract 3 from the current function. The equation becomes \[g(x) = (x - 2)^2 - 3\].

Step 4: Graphing the Functions

Graph the original function \(f(x) = x^2\) and the shifted function \(g(x) = (x - 2)^2 - 3\) on the same \(xy\)-plane. The graph of \(f(x) = x^2\) is a parabola opening upwards with vertex at (0, 0). The graph of \(g(x) = (x - 2)^2 - 3\) is the same parabola shifted right by 2 units and downward by 3 units, with vertex at (2, -3).

Key Concepts

Understanding Horizontal Shifts

When we talk about horizontal shifts in functions, we're essentially moving the function left or right along the x-axis. Imagine you have a parabola like the function \(f(x) = x^2\), which centers its vertex at the origin, point (0, 0). To achieve a horizontal shift, you adjust the input \(x\) of the function by adding or subtracting from it.

• If you want the function to move to the right, you replace \(x\) with \(x - h\), where \(h\) is the number of units you intend to shift. So, for \(h = 2\), our new function would be \(f(x) = (x - 2)^2\).
• Conversely, shifting the function to the left involves replacing \(x\) with \(x + h\). For example, \(f(x) = (x + 2)^2\) would shift the parabola to the left by 2 units.

This process changes the position of the vertex along the x-axis without affecting its shape.

Exploring Vertical Shifts

Vertical shifts in a function involve moving the function up or down along the y-axis. This adjustment directly adds or subtracts a value to the whole function output. For example, with the function \(f(x) = x^2\), we look at vertical shifts like so:

• To shift the function upwards, you add a positive constant \(k\). So, \(f(x) = x^2 + k\), where \(k\) represents how far we want to move up.
• To shift it downwards, subtract that constant: \(f(x) = x^2 - k\). In our case, for a downward shift of 3 units, the function becomes \((x - 2)^2 - 3\).

While the basic form of the graph remains as a parabola opening upwards, vertical shifts modify its position along the y-axis, effectively altering its vertex along the x-axis' path without deformation.

Graphing Parabolas

Graphing parabolas like \(f(x) = x^2\) is all about visualizing these shifts. Begin with the original parabolic function where the vertex sits at (0, 0) and it opens upwards from there. As you incorporate shifts:

• The horizontal shift acts first, repositioning the vertex sideways. For \(f(x) = (x-2)^2\), the vertex moves right to (2,0).
• Then, with the vertical shift, the whole parabola shifts up or down. Combining both shifts, our example becomes \(g(x) = (x-2)^2 - 3\), placing the vertex at (2, -3).

Once both shifts are applied, graph these transformations on the same \(xy\)-plane. You'll clearly see how the parabola has transformed from its original position to the new one, illustrating the power of function transformations.