Graph y=2 square root of x-x

Problem

y=2√(,x)−x

Solution

1. Determine the domain of the function. Since the expression contains √(,x) the value under the radical must be non-negative.

x≥0

2. Find the x-intercepts by setting y=0 and solving for x

0=2√(,x)−x

x=2√(,x)

x2=4*x

x*(x−4)=0

x=0,x=4

3. Find the derivative to determine the critical points and the slope of the function.

d(y)/d(x)=d(2*x(1/2)−x)/d(x)

d(y)/d(x)=x(−1/2)−1

d(y)/d(x)=1/√(,x)−1

4. Identify the maximum point by setting the derivative to zero.

1/√(,x)−1=0

√(,x)=1

x=1

5. Calculate the y-coordinate for the maximum point by substituting x=1 into the original equation.

y=2√(,1)−1

y=1

6. Analyze the behavior as x increases. As x→∞ the linear term −x dominates the square root term, so y→−∞

Final Answer

To graph y=2√(,x)−x plot the starting point (0,0) the maximum at (1,1) and the x-intercept at (4,0) then draw a curve that increases from the origin to the peak and decreases through the intercept toward negative infinity.

Want more problems? Check here!