Problem 53 - Calculus with Concepts in Calculus

Question

a. Let \(f(x)=x^{2}\) and \(g(x)=f(x+3)\). Using a suitable translation, sketch the graph of \(g\). b. Let \(f(x)=|x|\) and \(g(x)=f(x-2)\). Sketch the graph of \(g\)

Step-by-Step Solution

Answer

a. Shift parabola 3 units left. b. Shift V-shape 2 units right.

Step 1: Identify the Translation for Part (a)

The function \(f(x) = x^2\) is a simple quadratic function, which is a parabola opening upwards. For the function \(g(x) = f(x+3)\), we substitute \((x+3)\) into \(f(x)\), giving us \(g(x) = (x+3)^2\). This indicates a horizontal shift. Specifically, adding 3 inside the function translates the graph 3 units to the left.

Step 2: Sketch the Graph for Part (a)

Start with the graph of the basic function \(f(x) = x^2\) which is a parabola with its vertex at the origin (0,0). Since \(g(x) = (x+3)^2\) is a translation of \(f(x)\) 3 units to the left, draw a parabola identical in shape to \(f(x)\) but with its vertex at (-3,0).

Step 3: Identify the Translation for Part (b)

The function \(f(x) = |x|\) is a V-shaped graph with the vertex at the origin (0,0). The function \(g(x) = f(x-2)\) means substituting \((x-2)\) into \(f(x)\), resulting in \(g(x) = |x-2|\). This reflects a horizontal translation 2 units to the right.

Step 4: Sketch the Graph for Part (b)

Start with the basic V-shaped graph of \(f(x) = |x|\) with its vertex at (0, 0). For \(g(x) = |x-2|\), shift the entire graph 2 units to the right, resulting in the vertex being at (2, 0). The graph retains the same shape and orientation.

Key Concepts

Quadratic functions

Quadratic functions are polynomials of degree 2, represented by the standard form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The graph of a quadratic function is a parabola. Depending on the value of \(a\), the parabola opens upwards (if \(a > 0\)) or downwards (if \(a < 0\)).

Common properties of parabolas include:

• A vertex, which is the highest or lowest point of the parabola. In the function \(f(x) = x^2\), the vertex is at the origin \((0,0)\).
• Symmetry about a vertical axis that passes through the vertex.
• The parabola's width is determined by the coefficient \(a\). Larger values of \(|a|\) make the parabola narrower, while smaller values make it wider.

When transforming quadratic functions like moving them on the coordinate plane, we are often dealing with translations. For example, \(g(x) = (x+3)^2\) shifts the basic parabola, \(f(x) = x^2\), 3 units to the left without altering its shape.

Absolute value functions

Absolute value functions, such as \(f(x) = |x|\), are characterized by a V-shaped graph. This function creates a graph where it reflects any negative input value into a positive output, maintaining the essence of distance from zero.

Key attributes of absolute value functions include:

• The vertex, which is located at the origin \((0,0)\) for \(f(x) = |x|\).
• Two arms extending both upwards and forming a symmetric V-shape.
• The slope of each arm of the V is \(1\).

The graph of an absolute value function can be horizontally shifted by altering the input variable slightly. For instance, replacing \(x\) with \(x-2\) in \(f(x) = |x|\) results in \(g(x) = |x-2|\), which moves the vertex 2 units to the right, while maintaining the same V-shape and reflecting property.

Graph sketching

Graph sketching involves visually representing functions on a coordinate plane. This skill is crucial for understanding how functions behave and interact under various transformations.

To sketch graphs effectively:

• Identify the basic shape of the function, such as a parabola for quadratics or a V-shape for absolute values.
• Determine any transformations, like translations, which involve moving the graph horizontally or vertically without changing its shape.
• Pay attention to key points such as vertices, intercepts, and symmetry.

When sketching transformed graphs, like \(g(x) = (x+3)^2\) and \(g(x) = |x-2|\), it is helpful to understand how the changes inside the function affect the graph's position. Horizontal translations are guided by the change inside the parentheses. Thus, adding a number shifts the graph left, while subtracting shifts it right. Overall, mastering graph sketching empowers students to visualize complex algebraic concepts more intuitively.