The graph of which function will have a maximum and a $y$-intercept of 4?
A. $f(x)=-x^2+2 x+4$
B. $f(x)=4 x^2+6 x-1$
C. $f(x)=-4 x^2+8 x+5$
D. $f(x)=x^2+4 x-4$
Answer (1)
Check if the coefficient of x 2 is negative to determine if the function has a maximum.
Evaluate each function at x = 0 to find the y -intercept.
Identify the function that satisfies both conditions: having a maximum and a y -intercept of 4.
The function f ( x ) = − x 2 + 2 x + 4 has a maximum and a y -intercept of 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 that has a maximum and a y -intercept of 4. A quadratic function has a maximum if the coefficient of its x 2 term is negative. The y -intercept is the value of the function when x = 0 .
Analyzing Each Function Let's analyze each function:
f ( x ) = − x 2 + 2 x + 4 : The coefficient of x 2 is -1, which is negative, so it has a maximum. The y -intercept is f ( 0 ) = − 0 2 + 2 ( 0 ) + 4 = 4 .
f ( x ) = 4 x 2 + 6 x − 1 : The coefficient of x 2 is 4, which is positive, so it does not have a maximum. The y -intercept is f ( 0 ) = 4 ( 0 ) 2 + 6 ( 0 ) − 1 = − 1 .
f ( x ) = − 4 x 2 + 8 x + 5 : The coefficient of x 2 is -4, which is negative, so it has a maximum. The y -intercept is f ( 0 ) = − 4 ( 0 ) 2 + 8 ( 0 ) + 5 = 5 .
f ( x ) = x 2 + 4 x − 4 : The coefficient of x 2 is 1, which is positive, so it does not have 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 function with a maximum and a y -intercept of 4 is f ( x ) = − x 2 + 2 x + 4 .
Examples
Understanding quadratic functions and their properties, such as the y -intercept and the presence of a maximum or minimum, is crucial in various real-world applications. For instance, when modeling the trajectory of a projectile, the maximum height it reaches corresponds to the vertex of a downward-opening parabola (a quadratic function with a negative leading coefficient). The y -intercept can represent the initial height of the projectile. Similarly, in business, quadratic functions can model profit curves, where the maximum profit can be determined by finding the vertex of the parabola. Knowing the y -intercept can tell you the initial investment or fixed costs.
