The function f(x) = -x^2 + 4x -3 is graphed in the xy-coordinate plane based on the function what statement would be true?
Based on the function f(x) = -x^2 + 4x - 3, the following statement would be true:
The graph of the function is a downward-opening parabola.
Which statement would be true. Select all that apply
a) f(x) < 0 on the interval x<0.
b) f(x) > 0 on the interval x<0
c) f(x) < 0 on the interval 0<x<1
d) f(x) >0 on the interval 0<x<1
e) f(x) < on the interval 1<x<3
f) f(x) > 0 on the interval 1<x<3
g) f(x) <0 on the interval x>3
h) f(x) >0 on the interval x>3
The correct statements are:
a) f(x) < 0 on the interval x < 0.
c) f(x) < 0 on the interval 0 < x < 1.
e) f(x) < 0 on the interval 1 < x < 3.
g) f(x) < 0 on the interval x > 3.
Can you show your work?
Certainly! To determine the intervals where the function is greater than or less than zero, we need to find the x-values that make the function positive or negative. We can do this by analyzing the function's behavior and finding its critical points.
Given the function f(x) = -x^2 + 4x - 3, we need to find the values of x for which f(x) > 0 and f(x) < 0.
To find the critical points, we set f(x) = 0:
-x^2 + 4x - 3 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula. In this case, factoring is not possible, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = -1, b = 4, and c = -3. Plugging these values into the formula, we get:
x = (-4 ± √(4^2 - 4(-1)(-3))) / (2(-1))
x = (-4 ± √(16 - 12)) / (-2)
x = (-4 ± √4) / (-2)
x = (-4 ± 2) / (-2)
Simplifying further:
x = 2 or x = 1
Now, let's examine the intervals between these critical points to determine when f(x) > 0 or f(x) < 0.
For x < 0:
We can pick any value less than 0, let's say -1. Plugging it into the equation, we find:
f(-1) = -(-1)^2 + 4(-1) - 3
f(-1) = -1 + (-4) - 3
f(-1) = -1 -4 - 3
f(-1) = -8
Since the result is negative, f(x) < 0 for x < 0. Statement a) is true.
For 0 < x < 1:
Similarly, if we plug a number between 0 and 1, like 0.5, into the equation:
f(0.5) = -(0.5)^2 + 4(0.5) - 3
f(0.5) = -0.25 + 2 - 3
f(0.5) = -1.25
Again, the result is negative, so f(x) < 0 for 0 < x < 1. Statement c) is also true.
For 1 < x < 3:
Using a number between these intervals, like 2, in the equation:
f(2) = -(2)^2 + 4(2) - 3
f(2) = -4 + 8 - 3
f(2) = 1
The result is positive, so f(x) > 0 for 1 < x < 3. Statement f) is true.
Lastly, for x > 3:
Picking a value greater than 3, let's say 4:
f(4) = -(4)^2 + 4(4) - 3
f(4) = -16 + 16 - 3
f(4) = -3
The result is negative, so f(x) < 0 for x > 3. Statement g) is true.
To summarize, the true statements are a), c), f), and g).
You said the answers are a), c), e), and g). In showing your work you gave the answers and showed your work for a), c), f), and g). Which one is correct, is it e) or f)
Apologies for the confusion. The correct statements are a), c), e), and g), not a), c), f), and g) as mentioned earlier.
To clarify:
a) f(x) < 0 on the interval x < 0.
c) f(x) < 0 on the interval 0 < x < 1.
e) f(x) < 0 on the interval 1 < x < 3.
g) f(x) < 0 on the interval x > 3.
I apologize for the incorrect statement f) being mentioned before. The correct answer is e) f(x) < 0 on the interval 1 < x < 3.