Work each matching problem.
Match each equation in Column I with a description of its graph from Column II as it relates to the graph of y = x².
I II
a. y = (x - 7)² A. a translation to the left 7 units
b. y = x² - 7 B. a translation to the right 7 units
c. y = 7x² C. a translation up 7 units
d. y = (x + 7)² D. a translation down 7 units
e. y = x² + 7 E. a vertical stretching by a factor of 7

Verified step by step guidance

1. Step 1: Understand the basic transformations of the quadratic function y = x². These include translations (shifts) and stretches/compressions.
2. Step 2: For the equation y = (x - 7)², recognize that this represents a horizontal translation. Specifically, it shifts the graph of y = x² to the right by 7 units.
3. Step 3: For the equation y = x² - 7, identify this as a vertical translation. It shifts the graph of y = x² downward by 7 units.
4. Step 4: For the equation y = 7x², understand that this is a vertical stretch. The graph of y = x² is stretched vertically by a factor of 7.
5. Step 5: For the equation y = (x + 7)², recognize this as a horizontal translation. It shifts the graph of y = x² to the left by 7 units. Similarly, y = x² + 7 shifts the graph upward by 7 units.

Key Concepts

Transformations of Functions

Transformations of functions involve shifting, stretching, or reflecting the graph of a function. For quadratic functions like y = x², horizontal shifts occur when adding or subtracting a value from x, while vertical shifts happen when adding or subtracting a value from the entire function. Understanding these transformations is crucial for matching equations to their graphical representations.

Vertical Stretching and Compression

Vertical stretching and compression refer to how the graph of a function is altered by multiplying the function by a constant factor. For example, in the equation y = 7x², the graph is stretched vertically by a factor of 7, making it narrower compared to the standard parabola y = x². This concept is essential for recognizing how changes in the coefficient of x² affect the graph's shape.

Graphing Quadratic Functions

Graphing quadratic functions involves plotting parabolas, which are U-shaped curves. The standard form y = ax² + bx + c helps identify the vertex, axis of symmetry, and direction of opening. Recognizing how different forms of quadratic equations relate to the basic graph of y = x² allows for accurate matching of equations to their graphical descriptions.