To find the line of reflection, we need to find the equation of the line that the original function, f(x) = x^2 + 5x - 6, reflects over to result in f'(x) = -x^2 - 5x + 6.
The line of reflection will have the form x = a, where a is a constant.
To determine the value of a, we can compare corresponding points on the graph of f(x) and f'(x).
The vertex of f(x) is given by (h, k) = (-b/2a, f(-b/2a)). In this case, a = 1, b = 5, and c = -6.
Therefore, h = -5/(2*1) = -5/2 and k = f(-5/2) = (-5/2)^2 + 5(-5/2) - 6 = 11/4.
The vertex of f'(x) is given by (h', k') = (-b/2a, f'(-b/2a)). In this case, a = -1, b = -5, and c = 6.
Therefore, h' = -(-5)/(2*-1) = 5/2 and k' = f'(5/2) = -(5/2)^2 - 5(5/2) + 6 = 11/4.
Since the y-coordinates of the vertices of f(x) and f'(x) are the same (k = k'), the line of reflection must be the line x = h = -5/2.
Therefore, the equation for the line of reflection is x = -5/2.
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.
3 answers
what does y equal?
We can find the equation for the line of reflection by using the formula y = mx + b, where m is the slope of the line and b is the y-intercept.
To determine the slope of the line of reflection, we can consider two points on the line: one point on the original function f(x) and the corresponding reflected point on the reflected function f'(x).
Let's choose a point on f(x) that is not on the line of reflection. For simplicity, we can choose x = 0. Therefore, the corresponding y-coordinate can be found by substituting x = 0 into the equation f(x) = x^2 + 5x - 6:
f(0) = (0)^2 + 5(0) - 6 = -6.
Now, let's find the corresponding point on f'(x). Since the line of reflection is x = -5/2, we can evaluate f'(-5/2):
f'(-5/2) = (-5/2)^2 + 5(-5/2) + 6 = 11/4.
Therefore, the two points we have are (0, -6) and (-5/2, 11/4).
To find the slope, we can use the formula:
m = (y2 - y1) / (x2 - x1),
where (x1, y1) = (0, -6) and (x2, y2) = (-5/2, 11/4):
m = (11/4 - (-6)) / (-5/2 - 0)
= (11/4 + 24/4) / (-5/2)
= 35/4 / (-5/2)
= (35/4) * (-2/5)
= -14/4
= -7/2.
Now that we have the slope of the line, we need to find the y-intercept. We can use the point-slope form of the equation:
y - y1 = m(x - x1),
where (x1, y1) = (0, -6) and m = -7/2:
y - (-6) = (-7/2)(x - 0)
=> y + 6 = (-7/2)x
=> y = (-7/2)x - 6.
Therefore, the equation for the line of reflection is y = (-7/2)x - 6.
To determine the slope of the line of reflection, we can consider two points on the line: one point on the original function f(x) and the corresponding reflected point on the reflected function f'(x).
Let's choose a point on f(x) that is not on the line of reflection. For simplicity, we can choose x = 0. Therefore, the corresponding y-coordinate can be found by substituting x = 0 into the equation f(x) = x^2 + 5x - 6:
f(0) = (0)^2 + 5(0) - 6 = -6.
Now, let's find the corresponding point on f'(x). Since the line of reflection is x = -5/2, we can evaluate f'(-5/2):
f'(-5/2) = (-5/2)^2 + 5(-5/2) + 6 = 11/4.
Therefore, the two points we have are (0, -6) and (-5/2, 11/4).
To find the slope, we can use the formula:
m = (y2 - y1) / (x2 - x1),
where (x1, y1) = (0, -6) and (x2, y2) = (-5/2, 11/4):
m = (11/4 - (-6)) / (-5/2 - 0)
= (11/4 + 24/4) / (-5/2)
= 35/4 / (-5/2)
= (35/4) * (-2/5)
= -14/4
= -7/2.
Now that we have the slope of the line, we need to find the y-intercept. We can use the point-slope form of the equation:
y - y1 = m(x - x1),
where (x1, y1) = (0, -6) and m = -7/2:
y - (-6) = (-7/2)(x - 0)
=> y + 6 = (-7/2)x
=> y = (-7/2)x - 6.
Therefore, the equation for the line of reflection is y = (-7/2)x - 6.
1 answer