Problem 3 - 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. \(g(x)=(x+2)^{2}\)

Step-by-Step Solution

Answer

The graph of \(g(x)\) is obtained from the graph of \(f(x)\) by shifting the latter two units to the left.

Step 1: Identify the Transformation

Observe the given function \(g(x) = (x+2)^2\). This function demonstrates a shift of the \(x^2\) function to the left by two units. This can be understood as a horizontal translation from \(x\) to \(x+2\). There's no change in the function's scale or height.

Step 2: Sketch the Basic Function

Draw the graph of the basic function \(f(x) = x^2\), which is a parabola. Mark points for accuracy, such as at (0, 0), (-1,1), (1,1), (-2,4) and (2,4). This is intended to aid in drawing the final graph.

Step 3: Apply the Transformation

Translate every point on the graph \(f(x) = x^2\) to the left by two units to sketch \(g(x) = (x+2)^2\). As stated before, there is no vertical transformation.

Step 4: Verify the Transformation

Verify the sketch using a graphing utility. Plot the function \(g(x) = (x+2)^2\) on the graphing utility and compare the plotted graph with the manually sketched one. They should match!

Key Concepts

Horizontal Shifts

When we discuss horizontal shifts in functions, we're referring to moving the graph of a function left or right along the x-axis. This type of transformation changes only the x-values of the function without affecting its shape or size. For example, given a function like

• \( f(x) = x^2 \), the graph is a standard parabola centered at the origin (0,0).
• To create a horizontal shift, we adjust the variable inside the function. In the case of \( g(x)=(x+2)^2 \), the "+2" indicates a shift to the left by 2 units. This happens because we are effectively setting \( f(x) = x^2 \) equal to \( (x+2)^2 \).

Remember, a positive sign inside the brackets, \((x+c)\), moves the graph to the left, while a negative sign moves it right. Horizontal shifts maintain the original graph's characteristics like its orientation and symmetry.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, and they are often expressed in the standard form \( ax^2 + bx + c \). These functions produce a U-shaped graph known as a parabola. A basic quadratic function, like \( f(x) = x^2 \), is centered at the origin and is the most fundamental form of a parabola.

Some key features of quadratic functions include:

• The vertex, which is the parabola's highest or lowest point, depending on its orientation. For \( x^2 \), the vertex is at (0,0).
• The axis of symmetry, a vertical line that divides the parabola into two mirror-image halves. For \( x^2 \), the axis of symmetry is the y-axis.
• The direction in which the parabola opens, determined by the sign of \( a \). A positive \( a \) means the parabola opens upwards, while a negative \( a \) means it opens downwards.

Any transformation, like the horizontal shift we explored, does not alter these fundamental features. It simply relocates them on the coordinate plane.

Graphing Utilities

Graphing utilities, whether they are software applications, calculators, or online tools, are invaluable for visualizing mathematical functions and transformations. They help confirm manual sketches and ensure that the translations and transformations applied to a function are accurate. When working with functions like

• \( g(x) = (x+2)^2 \), using a graphing utility allows you to see exactly how the parabola shifts leftward by 2 units.
• These tools display graphs with precise measurement and scaling, eliminating human error often encountered in hand-drawn plots.

To use a graphing utility effectively:

• Input the transformed function and visually compare it with your hand-drawn sketch.
• Most utilities allow you to overlay multiple graphs, which aids in seeing the transformation clearly.
• Features can often be traced or highlighted to examine their positions, like vertex or intercepts.

Using these utilities not only aids in understanding transformations but also builds confidence in applying mathematical concepts correctly.