Consider the quadratic function [tex]$f(x)=\frac{1}{5} x^2-5 x+12$[/tex]. Which statements are true about the function and its graph? Select three options.
A. The value of [tex]$f(-10)=82$[/tex]
B. The graph of the function is a parabola.
C. The graph of the function opens down.
D. The graph contains the point [tex]$(20,-8)$[/tex].
E. The graph contains the point [tex]$(0,0)$[/tex].
Answer (2)
Evaluate f ( − 10 ) and confirm it equals 82: f ( − 10 ) = 82 .
Recognize that the graph of a quadratic function is a parabola.
Evaluate f ( 20 ) and confirm it equals -8: f ( 20 ) = − 8 .
Conclude that the three true statements are: f ( − 10 ) = 82 , the graph is a parabola, and the graph contains the point ( 20 , − 8 ) . f ( − 10 ) = 82 , parabola , ( 20 , − 8 )
Explanation
Analyzing the problem We are given the quadratic function f ( x ) = 5 1 x 2 − 5 x + 12 and asked to determine which of the given statements are true. Let's analyze each statement.
Checking statement 1 Statement 1: The value of f ( − 10 ) = 82 . We substitute x = − 10 into the function: f ( − 10 ) = 5 1 ( − 10 ) 2 − 5 ( − 10 ) + 12 = 5 1 ( 100 ) + 50 + 12 = 20 + 50 + 12 = 82 So, the first statement is true.
Checking statement 2 Statement 2: The graph of the function is a parabola. Since f ( x ) is a quadratic function (a polynomial of degree 2), its graph is indeed a parabola. So, the second statement is true.
Checking statement 3 Statement 3: The graph of the function opens down. The coefficient of the x 2 term is 5 1 , which is positive. Therefore, the parabola opens upwards, not downwards. So, the third statement is false.
Checking statement 4 Statement 4: The graph contains the point ( 20 , − 8 ) . We substitute x = 20 into the function: f ( 20 ) = 5 1 ( 20 ) 2 − 5 ( 20 ) + 12 = 5 1 ( 400 ) − 100 + 12 = 80 − 100 + 12 = − 20 + 12 = − 8 So, the fourth statement is true.
Checking statement 5 Statement 5: The graph contains the point ( 0 , 0 ) . We substitute x = 0 into the function: f ( 0 ) = 5 1 ( 0 ) 2 − 5 ( 0 ) + 12 = 0 − 0 + 12 = 12 Since f ( 0 ) = 12 = 0 , the fifth statement is false.
Final Answer Therefore, the three true statements are:
The value of f ( − 10 ) = 82
The graph of the function is a parabola.
The graph contains the point ( 20 , − 8 ) .
Examples
Understanding quadratic functions is crucial in various real-world applications. For instance, engineers use quadratic equations to model the trajectory of projectiles, such as rockets or balls. By analyzing the coefficients of the quadratic function, they can determine the maximum height, range, and other important parameters of the projectile's motion. Similarly, economists use quadratic functions to model cost and revenue curves, helping businesses optimize their production and pricing strategies. The vertex of the parabola represents the point of maximum profit or minimum cost, providing valuable insights for decision-making.
The true statements about the function f ( x ) = 5 1 x 2 − 5 x + 12 are: f ( − 10 ) = 82 , the graph is a parabola, and it contains the point ( 20 , − 8 ) . The graph opens upwards due to the positive leading coefficient. Thus, the selected options are A, B, and D.
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