Problem 42 - Calculus Single Variable

Question

Two functions \(f\) and \(g\) are given. Find a constant \(h\) such that \(g(x)=f(x+h)\). What horizontal translation of the graph of \(f\) results in the graph of \(g\) ? \(f(x)=x^{2}+4, g(x)=x^{2}-6 x+13\)

Step-by-Step Solution

Answer

The constant \(h\) is -3, which means a horizontal translation of 3 units to the left.

Step 1: Write Down Equations for g(x) and f(x+h)

We are given two functions: \(f(x) = x^2 + 4\) and \(g(x) = x^2 - 6x + 13\). We need to find a constant \(h\) such that \(g(x) = f(x+h)\), which translates the function \(f\) horizontally.

Step 2: Substitute f(x+h) in the Form g(x)

First express \(f(x+h)\) by replacing \(x\) with \(x+h\) in the function \(f(x)\). This gives us \(f(x+h) = (x+h)^2 + 4\).

Step 3: Expand f(x+h)

Expand \(f(x+h)\): \[f(x+h) = (x+h)^2 + 4 = x^2 + 2hx + h^2 + 4\].

Step 4: Set f(x+h) Equal to g(x)

Now, we create the equation \(x^2 + 2hx + h^2 + 4 = x^2 - 6x + 13\) by setting \(f(x+h) = g(x)\).

Step 5: Compare Coefficients and Solve for h

By comparing coefficients, we have:1. For coefficient of \(x\): \(2h = -6\)2. For the constant term: \(h^2 + 4 = 13\)Solving these equations, for the first equation \(2h = -6\), we get \(h = -3\). Substitute \(h = -3\) into the second equation: \((-3)^2 + 4 = 13\), which holds as \(9 + 4 = 13\). So, \(h = -3\) is correct.

Key Concepts

Function Transformation

In the realm of mathematics, function transformation refers to the modifications applied to a function's graph, altering its position or shape. These transformations include translations, reflections, dilations, and rotations. For our current focus, we delve into translations, specifically horizontal translations. In essence, a horizontal translation shifts the entire graph of a function left or right along the x-axis. This change is achieved by altering the input variable within the function. If a graph is shifted right by a certain number of units, the transformation is represented by replacing every instance of the variable, say \(x\), with \(x + h\), where \(h\) is a positive constant indicating the rightward shift. Conversely, a left shift is represented by \(x - h\), with \(h\) as a positive constant indicating the leftward movement. Understanding horizontal translations is crucial for visualizing how graphs can be manipulated to reflect changes in equations. This visualization discipline deepens comprehension of mathematical functions' dynamic nature. Keep in mind:

• Left shift: Replace \(x\) with \(x - h\)
• Right shift: Replace \(x\) with \(x + h\)

Quadratic Functions

Quadratic functions are a fundamental element of algebra, easily recognizable by their general form \(f(x) = ax^2 + bx + c\). This function features a distinct parabolic graph, which can either open upwards or downwards, depending on the sign of \(a\). In the case of \(a > 0\), the parabola opens upwards, exhibiting a 'U' shape. If \(a < 0\), the parabola flips to open downwards. Each quadratic function graph is symmetric concerning a vertical line called the axis of symmetry, which passes through the parabola's vertex. For quadratic functions in standard form, this axis is found using the formula \(x = -\frac{b}{2a}\). Another essential point associated with quadratic functions is the vertex, which represents the maximum or minimum point on the graph. The vertex's coordinates will provide insights into the function's range and are critical for understanding the function's behavior. Notably, these functions demonstrate how algebraic equations can represent various real-world scenarios, from physics' projectile motions to economics' cost analysis. Thus, understanding these concepts allows for graphical interpretations and aids in solving more complex equations.

Equation Solving

Solving equations is a crucial skill in mathematics, involving finding the values of variables that satisfy a given equation. In the context of function transformations and quadratic functions, solving equations often requires comparing different forms of expressions, as seen with horizontal translations. For instance, when we have two equations such as \(g(x) = x^2 - 6x + 13\) and \(f(x + h) = (x + h)^2 + 4\), our goal is to identify a value of \(h\) that aligns these functions. This process mandates expansion, simplification, and comparison of coefficients. By equating both sides of the equation, we break down the expression to solve for unknowns. One step involves expanding expressions like \((x + h)^2\) into \(x^2 + 2hx + h^2\) and then equating it to \(x^2 - 6x + 13\). The next key method is comparing coefficients, leading us to find the precise value of the transformation. In our example, solving \(2h = -6\) and \(h^2 + 4 = 13\) provides a systematic approach. Such techniques are not only pivotal in understanding mathematical theories but also in applying them to solve real-life problems.