Problem 24 - Beginning and Intermediate Algebra

Question

Sketch the graph of \(f(x) .\) Then, graph \(g(x)\) on the same axes using the transformation techniques. $$\begin{aligned}&f(x)=x^{2}\\\&g(x)=(x-3)^{2}\end{aligned}$$

Step-by-Step Solution

Answer

First, sketch the graph of \(f(x)=x^2\) using its key points (0,0), (1,1), (-1,1), (2,4), and (-2,4). Next, identify the transformation from \(f(x)\) to \(g(x)\) as a horizontal shift to the right by 3 units. Apply this transformation to the key points of \(f(x)\) to obtain the key points for \(g(x)\): (3,0), (4,1), (2,1), (5,4), and (1,4). Sketch the graph of \(g(x)=(x-3)^2\) using these key points on the same axes as \(f(x)\). Finally, label the graphs to distinguish \(f(x)\) and \(g(x)\).

Step 1: Sketch the graph of \(f(x) = x^2\)

To sketch the graph of \(f(x) = x^2\), recall that this is a standard parabolic function, with its vertex at the origin, opening upwards. To graph this function, plot the following key points and then connect them to form a smooth curve: - The vertex at (0,0) - The points (1,1) and (-1,1) - The points (2,4) and (-2,4)

Step 2: Understand the transformation from \(f(x)\) to \(g(x)\)

Now we need to determine how the function \(g(x) = (x-3)^2\) relates to \(f(x) = x^2\). We can see that \(g(x)\) is a result of a horizontal transformation of \(f(x)\); more specifically, it's a horizontal shift to the right by 3 units. This means that each point on the graph of \(f(x)\) will be shifted 3 units to the right to create the graph of \(g(x)\).

Step 3: Find the key points of \(g(x)\) from \(f(x)\) using the transformation

Taking the key points found in Step 1, we can apply the horizontal shift to find equivalent key points for the graph of \(g(x)\): - The vertex: (0 + 3, 0) = (3, 0) - The points: (1 + 3, 1) = (4, 1) and (-1 + 3, 1) = (2, 1) - The points: (2 + 3, 4) = (5, 4) and (-2 + 3, 4) = (1, 4)

Step 4: Sketch the graph of \(g(x)\) using the key points

Now that we have the key points for \(g(x)\), we can plot them on the same axes as the graph of \(f(x)\) and connect to form a smooth curve: - The vertex at (3,0) - The points (4,1) and (2,1) - The points (5,4) and (1,4)

Step 5: Final Step: Complete the graph with both functions

Sketch both the graph of \(f(x) = x^2\) and \(g(x) = (x-3)^2\) on the same set of axes using the key points and transformations found in the previous steps. Finally, label the graphs to distinguish them and make sure the transformations are visually accurate.

Key Concepts

Parabolic Function

A parabolic function is a special type of quadratic function that takes the shape of a parabola when graphed. Parabolas are symmetric and have a characteristic "U" shape.
The standard form of a parabolic function is expressed as:

• \(f(x) = ax^2 + bx + c\)

For the basic parabolic function \(f(x) = x^2\), the graph is a simple parabola that opens upwards, with its vertex at the origin (0,0).
Some key features of this parabolic graph include:

• Symmetrical about the vertical axis
• Vertex at the lowest point if the parabola opens upwards

The steepness of the parabolic curve and the exact position of the vertex can change when different coefficients are applied to the function. However, for \(f(x) = x^2\), its simplicity helps in understanding basic parabolic transformations.

Horizontal Shift

The horizontal shift is one type of transformation that can be applied to a graph, which moves the graph left or right along the x-axis.

In the transformation from \(f(x) = x^2\) to \(g(x) = (x - 3)^2\), a horizontal shift is performed. Here are the key points to note:

• The transformation \(x^2 \rightarrow (x-3)^2\) signifies moving each point of the graph 3 units to the right.
• If the function had been \(g(x) = (x + 3)^2\), each point would move 3 units to the left.
• Horizontal shifts do not affect the shape of the graph, only its position on the x-axis.

It's a simple yet powerful transformation to understand because it allows us to move graphs around without distorting their forms.

Vertex of a Parabola

The vertex of a parabola is a crucial point because it represents the highest or lowest point on the graph, depending on its orientation.
For a standard quadratic function \(f(x) = a(x - h)^2 + k\), the point (h, k) is the vertex.
In the parabolic function \(f(x) = x^2\), the vertex is located at the origin \( (0, 0) \).
For the transformed function \(g(x) = (x-3)^2\), the vertex shifts to:

• (3, 0) due to the horizontal shift of 3 units to the right.

The vertex gives you a starting point to sketch the rest of the graph and is essential in identifying the direction in which the parabola opens as well as its symmetry.

Quadratic Functions

Quadratic functions are polynomial functions with the highest degree of two, characterized by the general form:

• \(f(x) = ax^2 + bx + c\)

The graphs of quadratic functions are always parabolas. Depending on the sign of coefficient \a\:

• If \a > 0\, the parabola opens upwards.
• If \a < 0\, it opens downwards.

Quadratic functions can be transformed in various ways, including:

• Vertical shifts
• Horizontal shifts
• Reflecting over axes
• Stretching or compressing

These transformations can change the position and size of the parabola, but do not alter its fundamental "U" shape. Understanding these basics is crucial when dealing with more complex quadratic functions and their numerous transformations.