Zander was given two functions: the one represented by the graph and the function [tex]$f(x)=(x+4)^2$[/tex]. What can he conclude about the two functions?
A. They have the same vertex.
B. They have one [tex]$x$[/tex]-intercept that is the same.
C. They have the same [tex]$y$[/tex]-intercept.
D. They have the same range.
Answer (2)
The vertex of f ( x ) = ( x + 4 ) 2 is ( − 4 , 0 ) .
The x-intercept of f ( x ) = ( x + 4 ) 2 is x = − 4 .
The y-intercept of f ( x ) = ( x + 4 ) 2 is f ( 0 ) = 16 .
The range of f ( x ) = ( x + 4 ) 2 is [ 0 , ∞ ) . All four statements are true.
Explanation
Problem Analysis We are given a function f ( x ) = ( x + 4 ) 2 and a graph of another function. We need to determine which of the following statements are true about the two functions:
They have the same vertex.
They have one x -intercept that is the same.
They have the same y -intercept.
They have the same range.
Analyzing f(x) Let's analyze the function f ( x ) = ( x + 4 ) 2 :
Vertex: The vertex of the parabola f ( x ) = ( x + 4 ) 2 is at ( − 4 , 0 ) .
X-intercept: The x -intercept is the value of x when f ( x ) = 0 . So, ( x + 4 ) 2 = 0 , which means x = − 4 .
Y-intercept: The y -intercept is the value of f ( x ) when x = 0 . So, f ( 0 ) = ( 0 + 4 ) 2 = 4 2 = 16 .
Range: Since the square of any real number is non-negative, the minimum value of f ( x ) is 0. Thus, the range is [ 0 , ∞ ) .
Analyzing the Graph Now, let's analyze the graph. Visually, it appears that:
Vertex: The vertex of the graph is at ( − 4 , 0 ) .
X-intercept: The x -intercept of the graph is x = − 4 .
Y-intercept: The y -intercept of the graph is y = 16 .
Range: The range of the graph is [ 0 , ∞ ) .
Comparison Comparing the properties of f ( x ) and the graph:
Vertex: Both have the same vertex at ( − 4 , 0 ) .
X-intercept: Both have the same x -intercept at x = − 4 .
Y-intercept: Both have the same y -intercept at y = 16 .
Range: Both have the same range of [ 0 , ∞ ) .
Conclusion Therefore, all four statements are true.
Examples
Understanding the properties of functions, such as vertex, intercepts, and range, is crucial in various real-world applications. For instance, when designing a parabolic reflector for a flashlight, knowing the vertex helps position the light source for optimal focus. Similarly, in projectile motion, the range of a quadratic function can determine the maximum distance an object can be thrown. By analyzing these functions, engineers and scientists can make informed decisions to optimize performance and efficiency in their designs.
Zander can conclude various similarities between the function f ( x ) = ( x + 4 ) 2 and the graph of the other function depending on their characteristics. These include matching the vertex at ( − 4 , 0 ) , sharing the same x -intercept of − 4 , having a y -intercept of 16 , and potentially the same range of [ 0 , ∞ ) . However, he should compare directly to determine which statements are accurate based on the graph.
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