Graph $h(x)=0.5(x+2)^2-4$ by following these steps:
Step 1: Identify $a, h$, and $k$.
$a=0.5 \vee h=-2 \vee k=-4 \vee$
Step 2: Plot the vertex at $(-2,-4)$.
Step 3: The axis of symmetry is the line
$x=-2 v .$
Step 4: Evaluate the function at two other $x$-values:
$h(-4)=$
Answer (2)
Substitute x = − 4 into the function: h ( − 4 ) = 0.5 ( − 4 + 2 ) 2 − 4 .
Simplify inside the parenthesis: h ( − 4 ) = 0.5 ( − 2 ) 2 − 4 .
Calculate the square and multiply: h ( − 4 ) = 0.5 ( 4 ) − 4 = 2 − 4 .
Obtain the final result: h ( − 4 ) = − 2 .
Explanation
Understanding the problem The problem asks us to evaluate the function h ( x ) = 0.5 ( x + 2 ) 2 − 4 at x = − 4 . We are given the function and a specific value for x , and we need to find the corresponding value of h ( x ) .
Substituting x = -4 To find h ( − 4 ) , we substitute − 4 for x in the expression for h ( x ) : h ( − 4 ) = 0.5 ( − 4 + 2 ) 2 − 4
Simplifying the expression Now, we simplify the expression step by step: First, we evaluate the expression inside the parentheses: − 4 + 2 = − 2 So we have: h ( − 4 ) = 0.5 ( − 2 ) 2 − 4 Next, we square − 2 :
( − 2 ) 2 = 4 So we have: h ( − 4 ) = 0.5 ( 4 ) − 4 Now, we multiply 0.5 by 4 :
0.5 × 4 = 2 So we have: h ( − 4 ) = 2 − 4 Finally, we subtract 4 from 2 :
2 − 4 = − 2 Thus, h ( − 4 ) = − 2 .
Final Answer Therefore, the value of the function h ( x ) at x = − 4 is − 2 .
Examples
Understanding how to evaluate functions is essential in many real-world applications. For example, if h ( x ) represents the height of a ball thrown into the air at time x , then evaluating h ( − 4 ) would tell us the height of the ball 4 seconds before it was thrown. Similarly, in economics, if h ( x ) represents the cost of producing x items, then h ( − 4 ) might represent the cost savings from reducing production by 4 units. Evaluating functions helps us make predictions and understand relationships between variables in various fields.
To find h ( − 4 ) for the function h ( x ) = 0.5 ( x + 2 ) 2 − 4 , we substitute − 4 for x , simplify the equation step by step, and find that h ( − 4 ) = − 2 .
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