Plot the $x$-intercept of the function $f(x)=(x+4)^2$.
Answer (2)
Set the function equal to zero: ( x + 4 ) 2 = 0 .
Solve for x : x + 4 = 0 , which gives x = − 4 .
The x -intercept is the point where the function crosses the x -axis.
The x -intercept is − 4 .
Explanation
Understanding the Problem We are given the function f ( x ) = ( x + 4 ) 2 and asked to find and plot its x -intercept. The x -intercept is the point where the graph of the function intersects the x -axis. This occurs when f ( x ) = 0 .
Setting up the Equation To find the x -intercept, we set f ( x ) = 0 and solve for x :
( x + 4 ) 2 = 0
Solving for x Taking the square root of both sides, we get: x + 4 = 0
Finding the x-coordinate Subtracting 4 from both sides, we find: x = − 4
Identifying the x-intercept Therefore, the x -intercept is the point ( − 4 , 0 ) .
Final Answer The x -intercept of the function f ( x ) = ( x + 4 ) 2 is ( − 4 , 0 ) .
Examples
Understanding x-intercepts is crucial in various real-world applications. For instance, in physics, if f ( x ) represents the height of a projectile over time x , the x-intercept indicates when the projectile hits the ground. Similarly, in business, if f ( x ) represents profit as a function of sales x , the x-intercept shows the sales level needed to break even. Graphing and analyzing functions like this helps us visualize and understand key points in these scenarios, providing valuable insights for decision-making.
The x -intercept of the function f ( x ) = ( x + 4 ) 2 is found by setting the function to zero and solving for x . The solution gives us x = − 4 , meaning the x -intercept is the point ( − 4 , 0 ) . This indicates that the graph touches the x -axis at this point without crossing it, as it represents a double root.
;
