The graph of which function will have a maximum and a $y$-intercept of $4 ?$
A. $f(x)=4 x^2+6 x-1$
B. $f(x)=-4 x^2+8 x+5$
C. $f(x)=-x^2+2 x+4$
D. $f(x)=x^2+4 x-4
Answer (2)
Determine that a quadratic function has a maximum if the coefficient of the x 2 term is negative.
Find the y -intercept of each function by setting x = 0 .
Identify the function with a negative leading coefficient and a y -intercept of 4.
Conclude that the function is f ( x ) = − x 2 + 2 x + 4 , so the answer is f ( x ) = − x 2 + 2 x + 4 .
Explanation
Understanding the Problem We are given four quadratic functions and need to identify the one with a maximum and a y -intercept of 4. A quadratic function has a maximum if its leading coefficient (the coefficient of the x 2 term) is negative. The y -intercept is found by evaluating the function at x = 0 .
Analyzing Each Function Let's analyze each function:
f ( x ) = 4 x 2 + 6 x − 1 : The leading coefficient is 4, which is positive. Thus, this function has a minimum, not a maximum. The y -intercept is f ( 0 ) = 4 ( 0 ) 2 + 6 ( 0 ) − 1 = − 1 .
f ( x ) = − 4 x 2 + 8 x + 5 : The leading coefficient is -4, which is negative. Thus, this function has a maximum. The y -intercept is f ( 0 ) = − 4 ( 0 ) 2 + 8 ( 0 ) + 5 = 5 .
f ( x ) = − x 2 + 2 x + 4 : The leading coefficient is -1, which is negative. Thus, this function has a maximum. The y -intercept is f ( 0 ) = − ( 0 ) 2 + 2 ( 0 ) + 4 = 4 .
f ( x ) = x 2 + 4 x − 4 : The leading coefficient is 1, which is positive. Thus, this function has a minimum, not a maximum. The y -intercept is f ( 0 ) = ( 0 ) 2 + 4 ( 0 ) − 4 = − 4 .
Identifying the Correct Function From the analysis above, only f ( x ) = − x 2 + 2 x + 4 has a maximum and a y -intercept of 4.
Final Answer Therefore, the graph of the function f ( x ) = − x 2 + 2 x + 4 will have a maximum and a y -intercept of 4.
Examples
Understanding quadratic functions and their properties, like the y -intercept and the presence of a maximum or minimum, is crucial in various real-world applications. For instance, when designing a bridge, engineers use quadratic equations to model the arch's shape. The y -intercept can represent the starting point of the arch, and the maximum height ensures structural integrity. Similarly, in business, understanding the maximum profit a company can achieve based on production costs can be modeled using quadratic functions, where the y -intercept represents initial investment or fixed costs.
The function with a maximum and a y -intercept of 4 is f ( x ) = − x 2 + 2 x + 4 . This function has a negative leading coefficient, indicating a maximum, and evaluates to 4 when x = 0 . Therefore, the answer is option C.
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