Where does the graph of [tex]f(x)=-3 \sqrt{-2 x-3}[/tex] start?
A. [tex]\left(-\frac{3}{2}, 0\right)[/tex]
B. [tex]\left(\frac{3}{2},-3\right)[/tex]
C. [tex]\left(\frac{3}{2}, 3\right)[/tex]
D. [tex]\left(-\frac{3}{2}, 3\right)[/tex]
Answer (1)
Set the expression inside the square root to zero: − 2 x − 3 = 0 .
Solve for x : x = − 2 3 .
Substitute the value of x into the function f ( x ) to find the corresponding y value: f ( − 2 3 ) = 0 .
The starting point is ( − 2 3 , 0 ) .
Explanation
Understanding the Problem We are given the function f ( x ) = − 3 \t im es \t \t \t \t \t s q r t − 2 x − 3 and we want to find the starting point of its graph. The square root function is defined only for non-negative values. Thus, we need to find the value of x for which the expression inside the square root is equal to zero. This will give us the starting point of the graph.
Solving for x To find the starting point, we need to solve the equation − 2 x − 3 = 0 for x . Adding 3 to both sides gives − 2 x = 3 . Dividing both sides by -2, we get x = − 2 3 .
Finding the y-coordinate Now we need to find the value of the function f ( x ) at x = − 2 3 . Substituting this value into the function, we have
f ( − 2 3 ) = − 3 − 2 ( − 2 3 ) − 3 = − 3 3 − 3 = − 3 0 = − 3 ⋅ 0 = 0.
So, the y -coordinate of the starting point is 0.
The Starting Point Therefore, the starting point of the graph is ( − 2 3 , 0 ) .
Examples
Understanding the starting point of a square root function is crucial in various fields. For instance, in physics, when analyzing the motion of an object under a certain force, the square root function might describe the object's velocity. The starting point of the graph would then represent the initial velocity at a specific time. Similarly, in economics, a square root function could model the relationship between investment and profit, where the starting point indicates the minimum investment needed to start generating profit. Knowing this point helps in making informed decisions and predictions.
