$v(x)=\left\{\begin{array}{cl}-\frac{2}{7} x & \text { if } x<-1 \\\frac{7}{3} \sqrt[3]{x} & \text { if } x \geq-1\end{array}\right.
Step 1 of 3 : Identify the general shape and direction of the graph of this function on the interval ($-\infty,-1$).
Answer (1)
The function v ( x ) = − 7 2 x is a linear function on the interval ( − ∞ , − 1 ) .
The slope of the line is − 7 2 , which is negative.
A negative slope indicates that the line slopes downwards from left to right.
Therefore, the graph is a line sloping downwards from left to right on the interval ( − ∞ , − 1 ) .
Explanation
Identifying the Function Type We are given a piecewise function v ( x ) and asked to describe the shape and direction of its graph on the interval ( − ∞ , − 1 ) . On this interval, the function is defined as v ( x ) = − 7 2 x . This is a linear function, which means its graph is a straight line.
Determining the Direction To determine the direction of the line, we look at the coefficient of x , which is the slope of the line. In this case, the slope is − 7 2 . Since the slope is negative, the line slopes downwards from left to right. As x becomes more negative (i.e., moves towards − ∞ ), the value of v ( x ) increases.
Conclusion Therefore, on the interval ( − ∞ , − 1 ) , the graph of v ( x ) = − 7 2 x is a straight line sloping downwards from left to right.
Examples
Understanding the slope of a linear function is crucial in many real-world applications. For example, in physics, the velocity of an object decreasing at a constant rate can be modeled by a linear function with a negative slope. Similarly, in economics, a depreciation model where the value of an asset decreases linearly over time can be represented by a linear function with a negative slope. Knowing the slope helps predict how quickly the velocity decreases or how fast the asset loses value.
