Graph the function by plotting points.
f(x) = x²+3

To graph the function f ( x ) = x 2 + 3 , calculate the corresponding f(x) values for chosen x values from -3 to 3. Plotting the points ( − 3 , 12 ) , ( − 2 , 7 ) , ( − 1 , 4 ) , ( 0 , 3 ) , ( 1 , 4 ) , ( 2 , 7 ) , ( 3 , 12 ) will give you a U-shaped graph. Connecting the points reveals the quadratic nature of the function.
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Answered by Anonymous | 2025-07-04

Choose several x values: -3, -2, -1, 0, 1, 2, 3.
Calculate the corresponding f ( x ) values: f ( x ) = x 2 + 3 .
Determine the points: ( − 3 , 12 ) , ( − 2 , 7 ) , ( − 1 , 4 ) , ( 0 , 3 ) , ( 1 , 4 ) , ( 2 , 7 ) , ( 3 , 12 ) .
Plot the points and connect them to graph the function f ( x ) = x 2 + 3 .

Explanation

Understanding the Problem We are asked to graph the function f ( x ) = x 2 + 3 by plotting points. This means we need to choose some x values, calculate the corresponding f ( x ) values, and then plot the points ( x , f ( x )) on a coordinate plane.

Choosing x values Let's choose the following x values: -3, -2, -1, 0, 1, 2, 3. Now we will calculate the corresponding f ( x ) values for each x .

Calculating f(x) values For x = − 3 , we have f ( − 3 ) = ( − 3 ) 2 + 3 = 9 + 3 = 12 . So the point is ( − 3 , 12 ) .
For x = − 2 , we have f ( − 2 ) = ( − 2 ) 2 + 3 = 4 + 3 = 7 . So the point is ( − 2 , 7 ) .
For x = − 1 , we have f ( − 1 ) = ( − 1 ) 2 + 3 = 1 + 3 = 4 . So the point is ( − 1 , 4 ) .
For x = 0 , we have f ( 0 ) = ( 0 ) 2 + 3 = 0 + 3 = 3 . So the point is ( 0 , 3 ) .
For x = 1 , we have f ( 1 ) = ( 1 ) 2 + 3 = 1 + 3 = 4 . So the point is ( 1 , 4 ) .
For x = 2 , we have f ( 2 ) = ( 2 ) 2 + 3 = 4 + 3 = 7 . So the point is ( 2 , 7 ) .
For x = 3 , we have f ( 3 ) = ( 3 ) 2 + 3 = 9 + 3 = 12 . So the point is ( 3 , 12 ) .

Plotting the points Now we have the following points to plot: ( − 3 , 12 ) , ( − 2 , 7 ) , ( − 1 , 4 ) , ( 0 , 3 ) , ( 1 , 4 ) , ( 2 , 7 ) , ( 3 , 12 ) . Plotting these points and connecting them will give us the graph of the function f ( x ) = x 2 + 3 .


Examples
Understanding quadratic functions like f ( x ) = x 2 + 3 is crucial in many real-world applications. For instance, engineers use parabolas to design arches in bridges, ensuring structural stability. Similarly, the trajectory of a projectile, like a ball thrown in the air, follows a parabolic path, which can be modeled using a quadratic function. By analyzing the graph of such functions, one can determine key aspects such as the maximum height reached by the projectile or the optimal launch angle for maximum range. This knowledge is invaluable in fields ranging from sports to military applications.

Answered by GinnyAnswer | 2025-07-04