The function f(x)=x2+5x−6
is reflected over a given line and results in the equation f′(x)=−x2−5x+6
. Identify the equation for the line of reflection. (1 point)
The line of reflection is y=
.
AAAaannndd the bot gets it wrong yet again!
since f'(x) = -f(x), the line of reflection is y=0
Apologies for the mistake. You are correct. Since f'(x) = -f(x), the line of reflection is y = 0.
To identify the equation for the line of reflection, we need to find the line that is equidistant from the original function f(x) and its reflection f'(x).
The line of reflection will have the equation y = mx + b, where m represents the slope and b represents the y-intercept.
To find the slope of the line of reflection, we need to calculate the slope of the line passing through two corresponding points on the original function and its reflection.
Let's consider two points from the original function f(x):
Point A: (x1, f(x1))
Point B: (x2, f(x2))
And two corresponding points on the reflected function f'(x):
Point A': (x1, f'(x1))
Point B': (x2, f'(x2))
Since the line of reflection is equidistant from f(x) and f'(x), the slope of the line connecting A and A' should be equal to the slope of the line connecting B and B'.
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1) / (x2 - x1)
Let's calculate the slope for the points A and A' on f(x) and f'(x) respectively:
m1 = (f'(x1) - f(x1)) / (x1 - x1)
m2 = (f'(x2) - f(x2)) / (x2 - x2)
Since the slope of the line of reflection should be the same as the slopes of the two connecting lines, we have:
m1 = m2
Set the two equations equal to each other and solve for x:
(x1^2 + 5x1 - 6) - (x1^2 + 5x1 - 6) = -(x2^2 + 5x2 - 6) + (x2^2 + 5x2 - 6)
Simplify:
0 = -x2^2 - 5x2 + 6 + x2^2 + 5x2 - 6
Combine like terms:
0 = 0
Since this equation is true for all values of x, it means that any line with slope m can be the line of reflection.
Therefore, the equation for the line of reflection is y = mx + b, where m is the slope and b represents the y-intercept. However, we cannot determine the specific equation without more information.
The line of reflection can be found by equating the functions f(x) and f'(x) and solving for x:
f(x) = f'(x)
x^2 + 5x - 6 = -x^2 - 5x + 6
Rearranging the equation, we get:
2x^2 + 10x - 12 = 0
Dividing the equation by 2:
x^2 + 5x - 6 = 0
This quadratic equation can be factored as:
(x + 6)(x - 1) = 0
Setting each factor equal to zero, we find two possible values for x:
x + 6 = 0 or x - 1 = 0
Solving for x, we get:
x = -6 or x = 1
Therefore, the equation for the line of reflection is x = -6 and x = 1.