Problem 57 - A Graphical Approach to Precalculus with Limits

Question

Suppose that hand k are both positive numbers. Match each equation with the correct graph in choices A-D. (Graph cannot copy) $$y=(x+h)^{2}+k$$

Step-by-Step Solution

Answer

The correct graph is a parabola opening upward with vertex at \((-h, k)\).

Step 1 - Identify the Basic Form of the Function

The given equation is a quadratic function in the form \( y = (x + h)^2 + k \). This is a transformation of the standard quadratic function \( y = x^2 \).

Step 2 - Analyze Horizontal Translation

The term \( (x + h)^2 \) indicates a horizontal shift of the graph. The graph of \( y = x^2 \) is shifted to the left by \( h \) units when \( h \) is positive.

Step 3 - Analyze Vertical Translation

The term \( + k \) shifts the graph vertically. The graph of \( y = (x + h)^2 \) is shifted upwards by \( k \) units because \( k \) is positive.

Step 4 - Determine Vertex Position

The vertex of this parabola, originally at \( (0,0) \) in \( y = x^2 \), is moved to \( (-h, k) \) due to the horizontal \((h)\) and vertical \((k)\) transformations.

Step 5 - Sketch or Visualize the Graph

By applying the transformations, the new graph is a parabola opening upwards with its vertex at \( (-h, k) \). Based on graph choices, select the one matching this description.

Key Concepts

Graph Transformations

Graph transformations involve changing the position or shape of a graph relative to its original function. They are essential for understanding how equations and their graphs relate. In the context of quadratic functions, the basic equation is usually in the form of \( y = x^2 \). Transformation allows us to modify this graph either by shifting it across the Cartesian plane, stretching or compressing it, or reflecting it.

For a quadratic function like \( y = (x+h)^2 + k \), transformations tell us that the original graph of \( y = x^2 \) has been shifted according to the values of \( h \) and \( k \). Graph transformations keep the shape of the graph, a parabola in this case, but change its position.

Horizontal Translation

Horizontal translation involves moving the graph of a function left or right along the x-axis. In our quadratic function \( y = (x+h)^2 + k \), the term \( (x+h)^2 \) is crucial for this transformation.

When you see \( x + h \), it means that the graph of \( y = x^2 \) has been moved \( h \) units in the opposite direction of the sign next to \( h \). Thus, if \( h \) is positive, the graph shifts left by \( h \) units. Understanding this concept of opposite direction is essential for correctly positioning the graph.

This is different from other contexts where positive values might indicate a movement to the right.

Vertical Translation

Vertical translation shifts the graph up or down along the y-axis. It is dictated by the constant \( k \) in the equation \( y = (x+h)^2 + k \).

With vertical translation, you don't have to worry about reversing directions as you do with horizontal translation. A positive \( k \) means the graph moves up by \( k \) units, while a negative \( k \) would shift it down.

For our equation, since \( k \) is positive, the graph of \( y = x^2 \) will be elevated by \( k \) units. Vertical translation affects the entire graph consistently up or down without altering its shape.

Vertex of a Parabola

The vertex of a parabola is a pivotal point, marking its highest or lowest point, depending on the parabola's orientation. For a standard quadratic function \( y = x^2 \), the vertex is at the origin \((0,0)\).

In the transformed parabola \( y = (x+h)^2 + k \), the vertex becomes \((-h, k)\). The horizontal translation defined by \( h \) and the vertical translation defined by \( k \) determine this new position.

The vertex not only indicates location but also provides a clear view of how graph transformations affect the equation. In application, finding the vertex helps you sketch graphs or identify their position among options. This knowledge becomes very useful for graph analysis and real-world problem solving.