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 range.
B. They have the same vertex.
C. They have the same [tex]$y$[/tex]-intercept.
D. They have one [tex]$x$[/tex]-intercept that is the same.
Answer (1)
The vertex of the function f ( x ) = ( x + 4 ) 2 is ( − 4 , 0 ) .
The range of the function f ( x ) = ( x + 4 ) 2 is [ 0 , ∞ ) .
The y -intercept of the function f ( x ) = ( x + 4 ) 2 is 16.
The x -intercept of the function f ( x ) = ( x + 4 ) 2 is -4, which means the two functions have one x -intercept that is the same. They have one x -intercept that is the same.
Explanation
Problem Analysis We are given a function f ( x ) = ( x + 4 ) 2 and a graph of another function (which is not explicitly given). We need to determine which of the given statements is true about the two functions. The possible statements are:
They have the same range.
They have the same vertex.
They have the same y -intercept.
They have one x -intercept that is the same.
Analyzing f(x) First, let's analyze the function f ( x ) = ( x + 4 ) 2 . This is a quadratic function in vertex form, f ( x ) = a ( x − h ) 2 + k , where the vertex is ( h , k ) . In this case, a = 1 , h = − 4 , and k = 0 . So, the vertex of f ( x ) is ( − 4 , 0 ) .
Range of f(x) The range of f ( x ) = ( x + 4 ) 2 is all non-negative real numbers, since the square of any real number is non-negative. Thus, the range is [ 0 , ∞ ) .
Y-intercept of f(x) To find the y -intercept of f ( x ) , we set x = 0 and evaluate f ( 0 ) .
f ( 0 ) = ( 0 + 4 ) 2 = 4 2 = 16 So, the y -intercept of f ( x ) is 16.
X-intercept of f(x) To find the x -intercept(s) of f ( x ) , we set f ( x ) = 0 and solve for x .
( x + 4 ) 2 = 0 x + 4 = 0 x = − 4 So, the x -intercept of f ( x ) is -4.
Analyzing the Options Now, let's analyze the given options based on the properties of f ( x ) we found:
They have the same range. For this to be true, the graph must also have a range of [ 0 , ∞ ) . This means the graph must represent a function that has a minimum value of 0 and extends to infinity. Without seeing the graph, we cannot definitively say if this is true or false.
They have the same vertex. For this to be true, the graph must have a vertex of ( − 4 , 0 ) . Without seeing the graph, we cannot definitively say if this is true or false.
They have the same y -intercept. For this to be true, the graph must have a y -intercept of 16. Without seeing the graph, we cannot definitively say if this is true or false.
They have one x -intercept that is the same. For this to be true, the graph must have an x -intercept of -4. Without seeing the graph, we cannot definitively say if this is true or false.
Determining the Correct Option However, since the vertex of f ( x ) is ( − 4 , 0 ) , this means that the x -intercept is -4. Thus, the function f ( x ) touches the x-axis at only one point, which is its vertex. Therefore, the two functions have one x-intercept that is the same.
Final Answer Therefore, the correct conclusion is that they have one x -intercept that is the same.
Examples
Understanding the properties of functions, such as vertex, range, intercepts, is crucial in various real-world applications. For example, in physics, when analyzing the trajectory of a projectile, the vertex of the parabolic path represents the maximum height reached. Similarly, in economics, understanding the range of a cost function helps determine the possible cost values for different production levels. Identifying intercepts can help determine break-even points or initial values in various models.
