Problem 63 - 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. $$ h(x)-2(x-2)^{2}-1 $$

Step-by-Step Solution

Answer

The graph of the function \(h(x) = 2(x-2)^2 - 1\) is a parabola that opens upwards with its vertex at (2, -1) and is two times steeper than the standard quadratic function \(f(x) = x^2\).

Step 1: Plot the standard quadratic function

The standard quadratic function is given as \(f(x) = x^2\), which forms a parabola. This graph opens upwards and its vertex is at the origin, (0,0).

Step 2: Identify the transformations from the given function

The given function is \(h(x) = 2(x-2)^2 - 1\). Comparing it with the standard quadratic function, the transformations needed are: a horizontal shift to the right by 2 units (due to (x-2), a vertical shift downward by 1 unit (due to -1), and a vertical stretch by factor of 2 (due to the coefficient of 2).

Step 3: Apply the transformations to graph the function

Starting with the graph of \(f(x) = x^2\), move the curve to the right by 2 units, representing a horizontal shift. Then, move it downward by 1 unit to represent the vertical shift. Finally, stretch the resulting graph vertically by a factor of 2. The vertex of the new parabola is (2, -1), the graph opens upwards and it is 2 times steeper than the graph of the standard quadratic function.

Key Concepts

Graph Transformations

When working with quadratic functions, graph transformations are a powerful way to understand changes in a graph's appearance by applying shifts, stretches, or reflections. The standard form of a quadratic function is typically represented as \(f(x) = x^2\), which forms a simple, upward-opening parabola centered at the origin (0, 0). By applying specific transformations, we can modify the graph's position and shape without changing its basic parabolic nature.

These transformations include:

• Horizontal Shifts: Moving the graph left or right along the x-axis.
• Vertical Shifts: Moving the graph up or down along the y-axis.
• Vertical Stretches or Compressions: Making the graph appear taller or flatter.

Applying these concepts to the function \(h(x) = 2(x-2)^2 - 1\), we can see how transformations alter its shape and position compared to the standard quadratic function.

Parabola

The parabola is a U-shaped curve that graphically represents quadratic functions. For the standard quadratic function \(f(x) = x^2\), the parabola opens upward, indicating that its arms extend toward positive infinity as you move away from the vertex.

Every parabola has a distinctive set of features:

• Vertex: The highest or lowest point of the parabola.
• Axis of Symmetry: A vertical line that splits the parabola into two mirror images.
• Direction: Determines whether the parabola opens upwards or downwards, which depends on the sign of the coefficient before \(x^2\).

In graphing \(h(x) = 2(x-2)^2 - 1\), the parabola not only opens upward but also experiences transformations that change its position and steepness.

Vertex

The vertex of a parabola is a significant point that tells us a lot about the graph's orientation and position. In the parent function \(f(x) = x^2\), the vertex is at the origin (0, 0). It is located at the highest or lowest point on the graph, depending on the parabola's direction.

For the transformed function \(h(x) = 2(x-2)^2 - 1\), the vertex has moved due to the applied graph transformations:

• The horizontal shift to the right by 2 units results in the new vertex being at x = 2.
• The vertical shift downward by 1 unit changes the y-coordinate of the vertex to -1.

Thus, the vertex of \(h(x)\) is at (2, -1), which acts as the reference point for the location and shape of the graph.

Vertical Stretch

Vertical stretches alter the height of a parabola, making it narrower or wider. This transformation depends on the coefficient of the \(x^2\) term in the quadratic function. In the case of \(h(x) = 2(x-2)^2 - 1\), the coefficient 2 indicates a vertical stretch.

To visualize a vertical stretch:

• A coefficient greater than 1, like 2, makes the parabola narrower than the standard \(f(x) = x^2\) parabola.
• A coefficient between 0 and 1 would make it wider.

This stretching transformation means that for every x-value, the y-value of the graph is doubled compared to that of the standard parabola. This results in a steeper graph, emphasizing the height without altering the parabola's axis of symmetry.