Problem 44 - Calculus Single Variable

Question

In Exercises \(44-47,\) two functions \(f\) and \(g\) are given. Find constants \(h\) and \(k\) such that \(g(x)=f(x+h)+k .\) Describe the relationship between the plots of \(f\) and \(g\). \(f(x)=x^{2}, g(x)=x^{2}+2 x+5\)

Step-by-Step Solution

Answer

Constants are \( h = 1 \) and \( k = 4 \), translating \( f(x) \) left by 1 unit and up by 4 units.

Step 1: Express Both Functions

We need to express the given functions clearly. Here, we have function \( f(x) = x^2 \) and \( g(x) = x^2 + 2x + 5 \). Our goal is to rewrite \( g(x) \) in the form \( g(x) = f(x+h) + k \).

Step 2: Complete the Square for g(x)

Start with the given \( g(x) = x^2 + 2x + 5 \). To complete the square, find a number that makes \( x^2 + 2x \) a perfect square trinomial. Add and subtract \(1\): \( x^2 + 2x = (x + 1)^2 - 1 \). Thus, \( g(x) = (x + 1)^2 - 1 + 5 = (x + 1)^2 + 4 \).

Step 3: Identify h and k

Compare the squared form \( g(x) = (x + 1)^2 + 4 \) to the form \( g(x) = f(x + h) + k = (x + h)^2 + k \). Here, \((x + 1)^2\) implies \( h = 1 \) and the constant \(+4\) implies \( k = 4 \).

Step 4: Describe the Transformation

The constant \( h = 1 \) indicates a horizontal shift to the left by 1 unit, and \( k = 4 \) indicates a vertical shift upwards by 4 units. Therefore, the plot of \( g(x) \) is a translation of the plot of \( f(x) \) moved 1 unit left and 4 units up.

Key Concepts

Horizontal Shift

When we talk about horizontal shifts in the context of function transformations, we're referring to moving the graph of a function left or right along the x-axis. This shifting is accomplished without touching the shape of the original function's graph.

In our exercise, the transformation from function \( f(x) = x^2 \) to \( g(x) = x^2 + 2x + 5 \) involves such a shift. By "completing the square," we rewrote \( g(x) \) as \( (x + 1)^2 + 4 \). The term \((x + 1)^2\) indicates how the graph has been shifted horizontally. Specifically, it translates the graph one unit to the left. This is because the term \(x + 1\) inside the square can be viewed as \((x - (-1))\).

Here’s a quick way to think about it:

• If you have \((x - h)^2\), the function shifts \(h\) units to the right.
• If you have \((x + h)^2\), like in our example, the function shifts \(h\) units to the left.

Horizontal shifts are often a bit tricky for beginners because the direction seems reversed, but with practice, the logic becomes clear.

Vertical Shift

A vertical shift involves moving the entire graph of a function up or down on the y-axis without altering its shape. Such transformations affect only the range of the function, meaning they change the set of output values, but not how the x-values are processed.

In our exercise, the function \( g(x) = (x + 1)^2 + 4 \) includes a vertical shift. The constant term \(+4\) indicates that we move the parabola 4 units upwards.

Here's a simple idea to keep in mind:

• When you add a positive constant \(+k\) to \(f(x)\), the graph shifts \(k\) units up. For example, \(f(x) + 4\) shifts the function up by 4 units.
• Conversely, if you subtract \(-k\), as in \(f(x) - k\), it shifts down.

This transformation changes how high or low the graph sits, but again, the shape of the graph itself remains unchanged.

Completing the Square

The method of completing the square is an essential algebraic tool that helps simplify quadratic expressions. It also makes it easier to identify transformations such as shifts.

For our given function \( g(x) = x^2 + 2x + 5 \), completing the square meant finding a way to transform \(x^2 + 2x\) into a perfect square trinomial, which is easier to work with. Here's how we did it:

• First, we recognized \(x^2 + 2x\) could be transformed by adding and subtracting the square of half the coefficient of \(x\), which is \((2/2)^2 = 1\).
• We added and subtracted 1 to get \((x + 1)^2 - 1\).
• Our goal was to make \(x^2 + 2x\) into a perfect square, which becomes \((x + 1)^2\).

The final form, \(g(x) = (x + 1)^2 + 4\), tells us that \(g(x)\) is derived by shifting \(f(x)\) left by 1 and up by 4, magically achieving clarity through transformation.

Completing the square is not just for identifying shifts. It also helps in solving equations and analyzing the vertex form of quadratics, providing valuable insight into their properties.