Problem 5 - College Algebra and Calculus: An Applied Approach

Question

Describe the sequence of transformations from \(f(x)=x^{2}\) to \(g\). Then sketch the graph of \(g\) by hand. Verify with a graphing utility. \(g(x)=(x-2)^{2}+2\)

Step-by-Step Solution

Answer

The function \(g(x) = (x - 2)^{2} + 2\) is obtained from the base function \(f(x) = x^{2}\) by applying a horizontal shift to the right by 2 units and then a vertical shift upward by 2 units.

Step 1 - Identify the Transformations

The given function \(g(x) = (x - 2)^{2} + 2\) represents a parabola that is acquired by applying certain transformations to the base parabola \(f(x) = x^{2}\). Here: \n1. The term \((x - 2)\) inside the square function signifies a horizontal shift of the parabola \(f(x) = x^{2}\) to the right by 2 units. \n2. The '+2' outside the square function represents a vertical shift of the same parabola upward by 2 units. Hence, function \(g(x)\) can be obtained by first translating the function \(f(x)\) 2 units to the right and then 2 units up.

Step 2 - Sketching and Verifying the Graph

To sketch the graph of function \(g\), begin with the graph of \(f(x) = x^{2}\) which is a parabola with vertex at origin and facing upward. Then, shift this graph 2 units to the right and 2 units upwards. Now, the vertex of the parabola moved from the origin to the point (2,2). This is our graph \(g(x)\). Verification can then be done using a graphing utility by inputting the original \(f(x) = x^{2}\) and the transformed \(g(x) = (x - 2)^{2} + 2\) to check if the transformations were applied correctly.

Key Concepts

Horizontal Shift

When dealing with function transformations, a horizontal shift occurs when the graph of a function is moved left or right along the x-axis. In simple terms, the input to the function is modified. This can often be seen in expressions such as \(f(x - c)\), where the function shifts horizontally. If the \(c\) value is positive, the shift is to the right; if negative, it's to the left.

In our specific exercise, the function transformation goes from \(f(x) = x^2\) to \(g(x) = (x-2)^2 + 2\). Notice the part \(x-2\) inside the bracket. This indicates a horizontal shift of 2 units to the right because you're effectively "subtracting" 2 within the function's input.

• Shifts rightwards when you subtract a number from \x\.
• The graph of the function \(f(x)\) moves to \(x=2\) on the x-axis.

Understanding this helps in visualizing how and where the whole graph of a function moves horizontally across a coordinate plane.

Vertical Shift

A vertical shift impacts a graph by moving it up or down along the y-axis. This transformation affects the output of the function directly by modification outside the function expression, which could be spotting the presence of constants added or subtracted outside of the usual function form.

In our example, the transition from \(f(x) = x^2\) to \(g(x) = (x - 2)^2 + 2\) involves a vertical shift. The \(+2\) outside the squared function \((x-2)^2\) signifies that the graph of the parabola is lifted upwards by 2 units.

• Upward shift if a constant is added to the function.
• The graph of the function moves up along the y-axis to center at \(y=2\).

This type of shift changes the function's range but doesn't affect the x-value locations. It's essential for understanding how outputs are altered in relation to the original function.

Parabola Transformation

Parabolas are a well-known common type of function shaped like a symmetric curve or "U-shape" and represented usually by \(f(x) = x^2\) as their standard form. Understanding transformations helps in graphing these parabolas accurately as they undergo various manipulations.

In a function transformation challenge, like \(g(x) = (x - 2)^2 + 2\), it’s fundamental to grasp that such manipulations involve both shifting and repositioning vertices of the parabola.

• Starts with the basic structure \(y=x^2\).
• Horizontal shifts affect where the vertex lies regarding the x-axis.
• Vertical shifts impact the height of the vertex in relation to the y-axis.

Altogether, these shifts move the standard parabola's vertex from its original position at the origin \(0,0\) to a new location \(2,2\) aligning with any transformations we're applying. These manipulations essentially alter the vertex form of parabolas, where its equation modifies to \(a(x-h)^2 + k\) with \(h\) being the horizontal shift and \(k\) the vertical shift. Such understanding allows students to graph and predict transformations correctly, confirming results with graphing utilities to visualize and verify outcomes against calculations.