Expand the squared term: ( x − 5 ) 2 = x 2 − 10 x + 25 .
Substitute the expanded term back into the equation: y = ( x 2 − 10 x + 25 ) + 16 .
Combine the constant terms: y = x 2 − 10 x + 25 + 16 = x 2 − 10 x + 41 .
The standard form of the equation is y = x 2 − 10 x + 41 .
Explanation
Understanding the Problem The problem gives us the vertex form of a parabola's equation, y = ( x − 5 ) 2 + 16 , and asks us to find the equivalent standard form. The standard form of a quadratic equation is y = a x 2 + b x + c . To convert from vertex form to standard form, we need to expand and simplify the given equation.
Expanding the Squared Term First, let's expand the squared term ( x − 5 ) 2 . Recall that ( a − b ) 2 = a 2 − 2 ab + b 2 . Applying this, we get: ( x − 5 ) 2 = x 2 − 2 ( x ) ( 5 ) + 5 2 = x 2 − 10 x + 25
Substituting Back into the Equation Now, substitute this back into the original equation: y = ( x 2 − 10 x + 25 ) + 16
Simplifying the Equation Next, combine the constant terms: y = x 2 − 10 x + 25 + 16 = x 2 − 10 x + 41
Identifying the Correct Answer So, the standard form of the equation is y = x 2 − 10 x + 41 . Comparing this to the given options, we see that it matches option B.
Examples
Understanding quadratic equations is crucial in various real-world applications. For instance, engineers use parabolas to design arches in bridges, ensuring structural stability and optimal load distribution. Similarly, in physics, projectile motion follows a parabolic path, allowing us to predict the trajectory of objects like balls or rockets. By converting between vertex and standard forms, we can easily identify key features such as the vertex (highest or lowest point) and y-intercept, which are essential for these applications. For example, if we want to model the path of a ball thrown in the air, the vertex form helps us find the maximum height the ball reaches, while the standard form helps us determine where the ball will land.