f(x) = 5x - (4+7x)/3
Find two points on the line to graph the function.
Answer (2)
∙ Simplify the given function: f ( x ) = 3 8 x − 4 .
∙ Choose two x values: x 1 = 0 and x 2 = 1 .
∙ Calculate the corresponding y values: y 1 = − 3 4 and y 2 = 3 4 .
∙ The two points are ( 0 , − 3 4 ) and ( 1 , 3 4 ) .
Explanation
Understanding the Problem We are given the linear function f ( x ) = 5 x − 3 ( 4 + 7 x ) and we need to find two points on the line to graph it. To do this, we will choose two values for x , plug them into the function to find the corresponding y values, and then we will have two points ( x , y ) that we can use to graph the line.
Simplifying the Function First, let's simplify the function f ( x ) .
f ( x ) = 5 x − 3 4 + 7 x = 3 15 x − ( 4 + 7 x ) = 3 15 x − 4 − 7 x = 3 8 x − 4
So, f ( x ) = 3 8 x − 4 .
Calculating the Points Now, let's choose two values for x . A simple choice is x 1 = 0 and x 2 = 1 .
For x 1 = 0 , we have:
y 1 = f ( 0 ) = 3 8 ( 0 ) − 4 = 3 − 4 = − 3 4 ≈ − 1.33
So, the first point is ( 0 , − 3 4 ) .
For x 2 = 1 , we have:
y 2 = f ( 1 ) = 3 8 ( 1 ) − 4 = 3 8 − 4 = 3 4 ≈ 1.33
So, the second point is ( 1 , 3 4 ) .
Finding the Points Therefore, two points on the line are ( 0 , − 3 4 ) and ( 1 , 3 4 ) .
Examples
Linear functions are used in many real-world applications, such as calculating the cost of a taxi ride based on the distance traveled. The equation of the line can represent the relationship between the distance and the cost, and by finding two points on the line, we can easily graph the function and determine the cost for any given distance. Another example is calculating the speed of a car based on the time and distance traveled. The linear function can represent the relationship between time and distance, and by finding two points on the line, we can easily graph the function and determine the speed of the car.
We simplified the function f ( x ) = 5 x − 3 4 + 7 x to f ( x ) = 3 8 x − 4 . We then calculated two points by choosing x = 0 and x = 1 , resulting in points ( 0 , − 3 4 ) and ( 1 , 3 4 ) . These points can be plotted for graphing the line of the function.
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