Select the pair of points that both lie on the graph of the following function.
[tex]f(x)=3\left(\frac{7}{4} x+1\right)-\frac{9}{2} x[/tex]
A. $(0,3)$ and $(-4,0)$
B. $(3,0)$ and $(-4,0)$
C. $(3, 0)$ and $(0, -4)$
D. $(0,3)$ and $(0,-4)$
Answer (2)
First, simplify the function: f ( x ) = 4 3 x + 3 .
Then, check if the point ( 0 , 3 ) lies on the graph: f ( 0 ) = 3 .
Next, check if the point ( − 4 , 0 ) lies on the graph: f ( − 4 ) = 0 .
The pair of points that both lie on the graph of the function is ( 0 , 3 ) and ( − 4 , 0 ) .
Explanation
Simplify the function First, we need to simplify the given function: f ( x ) = 3 ( 4 7 x + 1 ) − 2 9 x f ( x ) = 4 21 x + 3 − 4 18 x f ( x ) = 4 3 x + 3
Check each pair of points Now, we will check each pair of points to see if they satisfy the simplified function.
Pair 1: ( 0 , 3 ) and ( − 4 , 0 ) For ( 0 , 3 ) :
f ( 0 ) = 4 3 ( 0 ) + 3 = 0 + 3 = 3 So, the point ( 0 , 3 ) lies on the graph. For ( − 4 , 0 ) :
f ( − 4 ) = 4 3 ( − 4 ) + 3 = − 3 + 3 = 0 So, the point ( − 4 , 0 ) lies on the graph. Since both points lie on the graph, this is the correct pair.
Pair 2: ( 3 , 0 ) and ( − 4 , 0 ) For ( 3 , 0 ) :
f ( 3 ) = 4 3 ( 3 ) + 3 = 4 9 + 3 = 4 9 + 4 12 = 4 21 = 5.25 Since f ( 3 ) = 5.25 = 0 , the point ( 3 , 0 ) does not lie on the graph.
Pair 3: ( 3 , 0 ) and ( 0 , − 4 ) Since we already know that ( 3 , 0 ) does not lie on the graph, we don't need to check ( 0 , − 4 ) .
Pair 4: ( 0 , 3 ) and ( 0 , − 4 ) Since x = 0 for both points, we can see that f ( 0 ) = 3 , so ( 0 , 3 ) lies on the graph, but ( 0 , − 4 ) does not.
Final Answer Therefore, the pair of points that both lie on the graph of the function is ( 0 , 3 ) and ( − 4 , 0 ) .
Examples
In real life, this type of problem can be used to determine if certain data points fit a linear model. For example, if you are tracking the growth of a plant over time, you can use a linear function to model the growth. By checking if the data points (time, height) lie on the graph of the function, you can validate the accuracy of your model. This is a fundamental concept in data analysis and modeling.
The pair of points that both lie on the graph of the function f ( x ) = 3 ( 4 7 x + 1 ) − 2 9 x is (0, 3) and (-4, 0) . This means option A is the correct answer. Both points satisfy the simplified function f ( x ) = 4 3 x + 3 .
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