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Understanding Graph Transformations in Quadratic Functions

This guide covers the transformations of quadratic graphs based on changes to their equations, specifically focusing on how shifting vertical positions affects function graphs. It analyzes the effects of translating graphs up or down, using examples like y = x² + 6 and y = x² + 2. Additionally, it explores real-world applications, such as solar energy collection using reflective surfaces, while determining the domain and range of associated quadratic functions. Practice questions included to reinforce understanding.

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Understanding Graph Transformations in Quadratic Functions

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  1. The graph would shift 2 units up. A The graph would shift 4 units up. B The graph would shift 4 units down. C The graph would shift 4 units to the left. D EXAMPLE 5 Standardized Test Practice How would the graph of the function y = x2 + 6 be affected if the function were changed to y = x2 + 2?

  2. ANSWER The correct answer is C. A D B C EXAMPLE 5 Standardized Test Practice SOLUTION The vertex of the graph of y = x2 + 6 is 6 units above the origin, or (0, 6). The vertex of the graph of y = x2 + 2 is 2 units above the origin, or (0, 2). Moving the vertex from (0, 6) to (0, 2) translates the graph 4 units down.

  3. EXAMPLE 6 Use a graph SOLAR ENERGY A solar trough has a reflective parabolic surface that is used to collect solar energy. The sun’s rays are reflected from the surface toward a pipe that carries water. The heated water produces steam that is used to produce electricity. The graph of the function y = 0.09x2 models the cross section of the reflective surface where xandyare measured in meters. Use the graph to find the domain and range of the function in this situation.

  4. EXAMPLE 6 Use a graph SOLUTION STEP 1 Find the domain. In the graph, the reflective surface extends 5 meters on either side of the origin. So, the domain is 5 ≤ x ≤ 5. STEP 2 Find the range using the fact that the lowest point on the reflective surface is (0, 0) and the highest point, 5, occurs at each end. y = 0.09(5)2= 2.25 Substitute 5 for x. Then simplify. The range is 0≤ y ≤ 2.25.

  5. Describe how the graph of the function y = x2+2 would be affected if the function were changed to y=x2– 2. WHAT IF? In Example 6, suppose the reflective surface extends just 4 meters on either side of the origin. Find the domain and range of the function in this situation. 7. 8. ANSWER ANSWER The graph would be translated 4 units down. Domain: – 4 ≤ x ≤ 4, Range: 0 ≤y ≤1.44 for Examples 5 and 6 GUIDED PRACTICE

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