the equation represents f(x) and the table shows some values of another quadratic function g(x)
f(x)=2(x^2-14+48)
x
-2,-1,0,1,2,3,4
g(x)
5, 0,-3,-4,-3, 0,5
select two of the statements that are correct about the given functions.
a. the line of symmmetry of f(x) is y=-2
b. the function g(x) has more x-intercepts than the function f(x)
c. the minimum value of f (x) is greater than the minimum value of g (x)
d. the x-value of the vertex of f (x) is greater than the x-value of the vertex of g(x)
b. the function g(x) has more x-intercepts than the function f(x)
c. the minimum value of f(x) is greater than the minimum value of g(x)
The given quadratic function f(x) = 2(x^2 - 14x + 48) can be simplified:
f(x) = 2(x^2 - 14x + 48)
= 2(x - 6)(x - 8)
= 2x^2 - 28x + 96
The x-values given in the table can be matched with the corresponding y-values of g(x):
x -2, -1, 0, 1, 2, 3, 4
g(x) 5, 0, -3, -4, -3, 0, 5
Now, let's check each statement:
a. The line of symmetry of f(x) is y = -2.
To find the line of symmetry of a quadratic function in the form ax^2 + bx + c, the formula is x = -b/2a.
In this case, the coefficient of x is -28 and the coefficient of x^2 is 2. Therefore, the line of symmetry is x = -(-28) / 2(2), which simplifies to x = 7.
Since this is the x-value of the vertex, the correct statement is:
a. The line of symmetry of f(x) is y = -2. (Incorrect)
b. The function g(x) has more x-intercepts than the function f(x).
To determine the x-intercepts, we need to find the x-values where the functions have a y-value of 0.
For f(x), we can set f(x) = 0 and solve for x:
2x^2 - 28x + 96 = 0
Using the quadratic formula, we find that f(x) has two x-intercepts.
For g(x), we can see from the table that g(x) has two x-intercepts as well.
Therefore, the correct statement is:
b. The function g(x) has more x-intercepts than the function f(x). (Incorrect)
c. The minimum value of f(x) is greater than the minimum value of g(x).
To find the minimum value of f(x), we can use the fact that the parabola opens upward because the coefficient of x^2 is positive. Since there is no coefficient on the x term in this case, the parabola has a vertex at the minimum point.
The x-value of the vertex can be found using the line of symmetry, which we found to be x = 7.
Substituting this value into f(x), we get:
f(7) = 2(7^2) - 28(7) + 96
= 98 - 196 + 96
= -2
Therefore, the minimum value of f(x) is -2.
For g(x), we can see from the table that it has a minimum value of -4.
Therefore, the correct statement is:
c. The minimum value of f(x) is greater than the minimum value of g(x). (Correct)
d. The x-value of the vertex of f(x) is greater than the x-value of the vertex of g(x).
As mentioned earlier, the x-value of the vertex of f(x) is 7.
For g(x), we can see from the table that it has a maximum value at x = 0, but the x-value of the vertex is not given.
Therefore, we cannot determine if the x-value of the vertex of f(x) is greater than the x-value of the vertex of g(x).
Therefore, the correct statement is:
d. The x-value of the vertex of f(x) is greater than the x-value of the vertex of g(x). (Incorrect)
In summary, the correct statements about the given functions are:
c. The minimum value of f(x) is greater than the minimum value of g(x).
To answer this question, let's analyze each statement:
a. The line of symmetry of a quadratic function is given by the equation x = -b/2a. Comparing this with the given function f(x) = 2(x^2 - 14x + 48), we can see that the coefficient of the term with x is -14. Therefore, the line of symmetry of f(x) can be found using the equation x = -(-14)/(2*2) = -14/4 = -7/2. So, statement a is incorrect.
b. To determine the number of x-intercepts for a quadratic function, we can look at the discriminant of the quadratic equation. The equation for g(x) is not explicitly provided, but from the given table, we can observe that g(x) passes through the x-axis (y = 0) at x = -1 and x = 3. Therefore, g(x) has 2 x-intercepts. On the other hand, the equation for f(x) is f(x) = 2x^2 - 28x + 96. This function is in vertex form, and we can see that the coefficient of x^2 is positive. Hence, f(x) does not intersect the x-axis and has no x-intercepts. Thus, statement b is correct.
c. To compare the minimum values of f(x) and g(x), we need to determine their vertex points. For a quadratic function in the form f(x) = ax^2 + bx + c, the x-value of the vertex can be found using the formula x = -b/2a, and the y-value (minimum value) of the vertex can be calculated by substituting the x-value into the function.
For f(x) = 2(x^2 - 14x + 48), the x-value of the vertex can be found using x = -(-14)/(2*2) = -14/4 = -7/2. To calculate the y-value of the vertex, substitute x = -7/2 into the function: f(-7/2) = 2(-7/2)^2 - 14(-7/2) + 48 = 2*49/4 + 49 + 48 = 98/4 + 49 + 48 = 24.5 + 49 + 48 = 121.5
Next, for g(x), we don't have the equation explicitly, but based on the table, we can see that the vertex occurs at x = 0, and the corresponding y-value is g(0) = -3.
Comparing the y-values, we find that f(x) has a minimum value of 121.5, whereas g(x) has a minimum value of -3. Therefore, statement c is incorrect.
d. From the previous calculations, we found that the x-value of the vertex for f(x) is -7/2, and for g(x) is 0. Hence, the x-value of the vertex of f(x) is less than the x-value of the vertex of g(x). Thus, statement d is incorrect.
Therefore, the correct statement out of the provided options is b. The function g(x) has more x-intercepts than the function f(x).