Problem 26 - Beginning and Intermediate Algebra

Question

Sketch the graph of \(f(x) .\) Then, graph \(g(x)\) on the same axes using the transformation techniques. $$\begin{array}{l}f(x)=x^{2} \\\g(x)=(x+1)^{2}\end{array}$$

Step-by-Step Solution

Answer

First, sketch f(x) = x^2 with its vertex at the origin (0,0). This graph is a basic upward-opening parabola. Next, find the vertex of g(x) = (x+1)^2 by noting the horizontal shift: g(x) is a shift of f(x) one unit to the left. So, the vertex of g(x) is at (-1,0). Sketch g(x) on the same axes as f(x) with the same shape as f(x) and compare the two graphs. Both are upward opening parabolas, with g(x) shifted one unit left compared to f(x).

Step 1: Sketch the graph of f(x)

To start, we need to find the vertex of the parabola given by f(x) = x^2. The vertex of f(x) can be found using the formula \(h = \frac{-b}{2a}\) and \(k = f(h)\). For f(x) = x^2, a = 1 and b = 0. So the vertex is: \[ h = \frac{-0}{2\cdot1} = 0 \] \[ k = f(h) = f(0) = 0^2 = 0 \] So, the vertex of the parabola f(x) = x^2 is at the point (0, 0). Next, we'll sketch the basic shape of f(x) = x^2 with its vertex at the origin. Remember that quadratic functions with a positive leading coefficient (a = 1 in this case) open upward, creating a U-shaped graph.

Step 2: Apply transformation to f(x) to obtain g(x)

Now, we'll focus on transforming f(x) into g(x) = (x+1)^2. Notice that g(x) is a horizontal shift of f(x) by one unit to the left. We can re-write g(x) as follows: \[ g(x) = (x-(-1))^2 \] Using this form, we can identify that the vertex of g(x) is shifted one unit to the left from the vertex of f(x). So, the vertex of g(x) would be at the point (-1, 0).

Step 3: Sketch g(x) on the same axes as f(x)

Using the information we found in the previous steps, we can now sketch g(x) on the same axes as f(x). Since g(x) is just a horizontal shift of f(x) to the left, the graph of g(x) will have the same shape as f(x), with the vertex at the point (-1, 0). This is the only difference between the two graphs. Once both functions are plotted on the same axes, compare and analyze the graphs. They are both upward opening parabolas with the same shape, but g(x) is shifted one unit to the left compared to f(x).

Key Concepts

Parabola Vertex

The vertex of a parabola is a key feature that helps us to understand its graph. In simpler terms, the vertex is the "turning point" of the parabola. For the quadratic function \(f(x) = ax^2 + bx + c\), the vertex can be found using the formulas for the coordinates as follows:

• \(h = \frac{-b}{2a}\)
• \(k = f(h)\)

If we plug these into \(f(x) = x^2\), where \(a = 1\) and \(b = 0\), the vertex is at \((0, 0)\). This is crucial because it represents the most commonly known parabolic form—a vertex at the origin with a U-shape opening.
The vertex not only locates the parabola on the graph but also tells us whether the parabola opens upwards or downwards, depending on the sign of \(a\). If \(a\) is positive, like in \(f(x) = x^2\), the graph opens upward.

Graph Transformation

Graph transformation is a powerful tool that helps us create variations of a function. By using transformations, we can modify the graph's shape, location, and orientation without altering its basic form.
For quadratic functions, transformations include shifts, stretches, compressions, and reflections. These help in visualizing how changes in the equation affect the graph:

• Shifting the graph along the x-axis or y-axis
• Stretching or compressing to alter the steepness
• Reflecting over the axes

In our case, transforming \(f(x) = x^2\) to \(g(x) = (x+1)^2\) mainly involves a horizontal shift. The transformation maintains the "U" shape, but its vertex moves.
Understanding these transformations is fundamental in graphing because it allows flexibility and creativity in managing how functions are represented visually.

Horizontal Shift

The horizontal shift is among the simplest transformations, involving moving the graph left or right. This shift does not affect the y-coordinates of points on the graph; it only affects their x-coordinates.
To perform a horizontal shift, we modify the \(x\) term inside the function. The function \(g(x) = (x+1)^2\) is a perfect example of a horizontal shift applied to \(f(x) = x^2\):

• The term \(\(+1\)\) inside the brackets \((x+1)^2\) indicates a shift one unit to the left.

Therefore, the vertex moves from \((0,0)\) in \(f(x)\) to \((-1,0)\) in \(g(x)\). This change occurs because the addition moves each point of the graph to the left, reflecting the leftward horizontal shift.
The crucial point to remember here is that a change inside the function (with the \(x\)-variable) shifts the graph horizontally, and its direction depends on the sign: within \((x + 1)\), moving left, or \((x - 1)\), moving right.