Question

Functions 1 and 2 are shown:

Function 1: f(x) = −3x2 + 4x + 2

Function 2. A graph of a parabola that opens down that goes through points negative one half comma 0, one half comma 4, and 2 comma 0 is shown.

Which function has a larger maximum?

a
Function 1 has a larger maximum.

b
Function 2 has a larger maximum.

c
Function 1 and Function 2 have the same maximum.

d
Function 1 does not have a maximum value.

Answer 1

To determine which function has a larger maximum, we need to compare the vertex points of both functions.

For Function 1, the equation is f(x) = -3x^2 + 4x + 2. The coefficient of the x^2 term is negative, indicating that the parabola opens downward. The vertex of the parabola can be found using the formula x = -b / (2a), where a is the coefficient of x^2 (-3) and b is the coefficient of x (4).

Using the formula, we find that the x-coordinate of the vertex is x = -4 / (2 * -3) = 2/3. Substituting this value into the equation, we can find the corresponding y-coordinate: f(2/3) = -3(2/3)^2 + 4(2/3) + 2 = 4/3.

For Function 2, the given information states that the graph is a parabola that opens downward and passes through the points (-1/2, 0), (1/2, 4), and (2, 0). The vertex of this parabola can be found as the midpoint between the x-coordinates of the two known points with equal y-values. In this case, it would be the midpoint between (-1/2, 0) and (2, 0), which is (3/4, 0). Therefore, the maximum value is y = 0.

Comparing the y-values of the vertices, we see that f(2/3) = 4/3 and f(3/4) = 0. Since 4/3 is greater than 0, we can conclude that Function 1 has a larger maximum.

Therefore, the correct answer is:

a) Function 1 has a larger maximum.

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