What are the two equations created by the inequality $|x-12|+5<27$? $y_1=\sqrt{|x-12|}$ and $y_2=\sqrt{22}$ Complete which graph represents the two functions?
Answer (2)
The inequality ∣ x − 12∣ + 5 < 27 leads to the range − 10 < x < 34 . The functions y 1 = ∣ x − 12∣ and y 2 = 22 intersect at points ( − 10 , 22 ) and ( 34 , 22 ) . The graph visually represents these relationships, showing the square root function's vertex and the constant line.
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The function y 1 = ∣ x − 12∣ is a square root function with vertex at x = 12 .
The function y 2 = 22 is a horizontal line at y = 22 .
The intersection points are found by solving ∣ x − 12∣ = 22 , giving x = − 10 and x = 34 .
The graph shows a square root function opening to the left and right, intersected by a horizontal line at y = 22 .
Explanation
Analyze the functions We are given two functions: y 1 = ∣ x − 12∣ and y 2 = 22 . We need to determine which graph represents these two functions. The function y 1 involves an absolute value inside a square root, which means it will be symmetric around the vertex. The function y 2 is a constant function, which will be a horizontal line.
Determine the shape of the graphs The function y 1 = ∣ x − 12∣ is a square root function with a vertex at x = 12 . For x < 12 , the function is 12 − x and for 12"> x > 12 , the function is x − 12 . The graph starts at ( 12 , 0 ) and increases as we move away from x = 12 in either direction. The function y 2 = 22 is a horizontal line at y = 22 ≈ 4.69 .
Find the intersection points To find the intersection points of the two graphs, we set ∣ x − 12∣ = 22 . Squaring both sides, we get ∣ x − 12∣ = 22 . This gives us two equations: x − 12 = 22 or x − 12 = − 22 . Solving for x , we get x = 34 or x = − 10 . The intersection points are ( − 10 , 22 ) and ( 34 , 22 ) .
Conclusion Based on the analysis, the graph should show a square root function y 1 = ∣ x − 12∣ that opens to the left and right from the vertex at ( 12 , 0 ) , and a horizontal line y 2 = 22 intersecting the square root function at x = − 10 and x = 34 .
Examples
Understanding and graphing functions like these can help in various real-world scenarios. For example, in physics, the velocity of an object might be modeled by a square root function, and a constant function could represent a constant force acting on the object. Finding the intersection points helps determine when the velocity reaches a certain level under the influence of that force. Similarly, in engineering, these functions could model the stress on a material, and the intersection points could indicate when the material reaches its breaking point.
