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. Identify the base graph: The parent function is \(y = x^{2}\), which is a parabola centered at the origin.
2. Understand horizontal translations: For equations of the form \(y = (x - h)^{2}\), the graph shifts horizontally. If \(h\) is positive, the graph moves right by \(h\) units; if \(h\) is negative, it moves left by \(|h|\) units.
3. Understand vertical translations: For equations of the form \(y = x^{2} + k\), the graph shifts vertically. If \(k\) is positive, the graph moves up by \(k\) units; if \(k\) is negative, it moves down by \(|k|\) units.
4. Understand vertical stretching: For equations of the form \(y = a x^{2}\), where \(a > 1\), the graph is vertically stretched by a factor of \(a\), making it narrower.
5. Match each equation to its description by applying these rules: (a) \(y = (x - 7)^{2}\) shifts right 7 units, (b) \(y = x^{2} - 7\) shifts down 7 units, (c) \(y = 7 x^{2}\) is a vertical stretch by factor 7, (d) \(y = (x + 7)^{2}\) shifts left 7 units, and (e) \(y = x^{2} + 7\) shifts up 7 units.

Key Concepts

Horizontal Translations of Quadratic Functions

Horizontal translations shift the graph of a function left or right without changing its shape. For y = (x - h)², the graph moves h units to the right if h is positive, and for y = (x + h)², it moves h units to the left. This shift affects the vertex's x-coordinate but not the parabola's width or direction.

Vertical Translations of Quadratic Functions

Vertical translations move the graph up or down by adding or subtracting a constant outside the squared term. For y = x² + k, the graph shifts up k units if k is positive, and down k units if k is negative. This changes the vertex's y-coordinate but keeps the parabola's shape and orientation intact.

Vertical Stretching of Quadratic Functions

Vertical stretching changes the width and steepness of the parabola by multiplying the squared term by a factor a. For y = a x², if |a| > 1, the graph becomes narrower (stretched vertically), and if 0 < |a| < 1, it becomes wider (compressed). The vertex remains at the origin if no translations occur.