Problem 57 - College Algebra

Question

Begin by graphing the standard quadratic function, \(f(x)-x^{2} .\) Then use transformations of this graph to graph the given function. $$ h(x)--(x-2)^{2} $$

Step-by-Step Solution

Answer

The graph of the function \(h(x) = (x-2)^{2}\) is a parabola opening upwards with its vertex at (2, 0) and the axis of symmetry along the vertical line x=2. This is obtained by shifting the graph of the standard quadratic function \(f(x) = x^{2}\) two units to the right.

Step 1: Graph the Standard Quadratic Function (SQF)

Start with the graph of the SQF, \(f(x)=x^{2}\). This is a parabola that opens upwards with its vertex at the origin (0, 0) and the axis of symmetry along the y-axis.

Step 2: Analyze the Given Function

Now move to the given function \(h(x)=(x-2)^{2}\). Comparing this with the SQF, it can be noticed that the 'x' in the SQF has been replaced with '(x-2)'. This is a horizontal shift (or translation) of the SQF 2 units to the right.

Step 3: Apply the Transformation

Applying this transformation to the graph of the SQF gives the graph of \(h(x)=(x-2)^{2}\). This is a parabola that opens upwards like the SQF, but its vertex is at (2,0) and the axis of symmetry is the vertical line x=2.

Key Concepts

Quadratic Transformations

Quadratic transformations involve modifying the basic graph of a quadratic function, often written as \(f(x) = x^2\), to create a new graph. This process can include various changes, such as moving or reshaping the parabola.
For instance, any change to the standard form \(f(x) = x^2\) is considered a transformation. Adjusting parameters like the coefficient before the \(x^2\), terms added or subtracted to the \(x\), or the whole function, changes the graph. These modifications are known as transformations.

• Vertical transformations affect the parabola's height or position on the y-axis.
• Horizontal transformations affect the size or position on the x-axis, shifting where the graph lies horizontally.

Understanding how these transformations work helps in predicting the appearance and position of a quadratic graph.

Horizontal Shift

A horizontal shift is a type of transformation where the entire graph of a function moves left or right along the x-axis. This does not change the shape of the graph or its orientation.
When you see a function like \(h(x) = (x-2)^2\), the \(-2\) indicates a movement to the right. To visualize this:

• The parent function, \(f(x) = x^2\), has its vertex at (0, 0).
• With \(x-2\) as the argument, each point moves 2 units right.

This horizontal shift causes the vertex of the parabola to move from the origin to the point (2, 0).
It's important because altering \(x\) directly impacts where the function values begin to change. Every x-coordinate becomes \(x + 2\). Always remember, replacing \(x\) with \(x-h\) translates the graph by \(h\) units to the right.

Vertex of Parabola

The vertex of a parabola is the point where it changes direction, and it can be thought of as the "tip" of the curve. In the standard quadratic form \(y = ax^2 + bx + c\), finding the vertex involves using a formula when given transformations.
For any parabola \(f(x) = (x-h)^2 + k\), the vertex shifts from its origin (0, 0) to the coordinates \((h, k)\). This is crucial since it determines the minimum or maximum value of the function.
Consider the function \(h(x) = (x-2)^2\):

• It is derived from the basic \(x^2\) with a transformation of \(x\), thus shifting the vertex from (0,0) to (2,0).
• The parameter \(h = 2\) shifts the graph horizontally.

Knowing where the vertex is allows us to draw accurate graphs and understand changes in parabola behavior.

Axis of Symmetry

The axis of symmetry for a parabola is a vertical line that divides the graph into two identical halves. It's symbolic of the balance and reflective properties of the graph.
This line runs through the vertex and is key in graphing, alongside the vertex itself. For a function \(y = a(x-h)^2 + k\), the axis of symmetry is given by \(x = h\). In the exercise \(h(x) = (x-2)^2\):

• The vertex is at (2, 0), making the axis of symmetry \(x = 2\).
• Every point on the parabola seems to mirror around this vertical line.

This concept simplifies checking for errors in graphing and helps in understanding features of the parabola, like its width and direction. Knowing the axis of symmetry is crucial because it ensures that the graph is centered and accurately represents the quadratic function.