Graph this function: [tex]f(x)=\frac{3}{4} x-2[/tex]
Answer (2)
Find the y-intercept by setting x = 0 : f ( 0 ) = − 2 , so the y-intercept is ( 0 , − 2 ) .
Find another point on the line by choosing a value for x , such as x = 4 : f ( 4 ) = 1 , so another point is ( 4 , 1 ) .
Plot the points ( 0 , − 2 ) and ( 4 , 1 ) on the graph.
Draw a straight line through these two points to represent the function f ( x ) = 4 3 x − 2 .
Explanation
Understanding the Problem We are asked to graph the function f ( x ) = 4 3 x − 2 . This is a linear function, so we need to find two points on the line to graph it.
Finding the y-intercept First, let's find the y-intercept. The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0 . Plugging in x = 0 into the function, we get f ( 0 ) = 4 3 ( 0 ) − 2 = − 2 . So the y-intercept is the point ( 0 , − 2 ) .
Finding another point on the line Next, let's find another point on the line. We can choose any value for x and plug it into the function to find the corresponding y value. Let's choose x = 4 . Plugging in x = 4 into the function, we get f ( 4 ) = 4 3 ( 4 ) − 2 = 3 − 2 = 1 . So another point on the line is ( 4 , 1 ) .
Graphing the line Now we have two points on the line: ( 0 , − 2 ) and ( 4 , 1 ) . We can plot these two points on the graph and draw a straight line through them. This line represents the graph of the function f ( x ) = 4 3 x − 2 .
Final Answer The graph of the function f ( x ) = 4 3 x − 2 is a straight line that passes through the points ( 0 , − 2 ) and ( 4 , 1 ) .
Examples
Understanding linear functions like f ( x ) = 4 3 x − 2 is crucial in many real-world applications. For instance, imagine you're tracking the distance a train travels over time. If the train moves at a constant speed, the relationship between time and distance can be modeled by a linear function. The slope represents the speed of the train, and the y-intercept could represent the starting point. Graphing this function allows you to quickly visualize the train's position at any given time, making it easier to plan schedules and estimate arrival times. Similarly, linear functions are used in economics to model supply and demand curves, helping businesses make informed decisions about pricing and production.
To graph the function f ( x ) = 4 3 x − 2 , identify the y-intercept at ( 0 , − 2 ) and another point at ( 4 , 1 ) . Plot these points and draw a straight line through them. The slope of the function is 4 3 , indicating a consistent relationship between the values of x and f ( x ) .
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