Answer:
The point (2 - d, y) is on the graph of f(x)
The line x = 2 is the axis of symmetry of the graph of f(x)
Explanation:
* Lets explain how to prove that a point lies on a graph Algebraically
- Substitute the value of the x-coordinate of the point in the equation
of the graph the answer must be equal the y-coordinate of the point
- The function is a quadratic because the greatest power of x is 2,
then it represented by parabola
- The parabola has a vertex point (h , k), where h is the x-coordinate
and k is the y-coordinate
- This vertex divides the parabola into two equal parts, then the axis
of symmetry of the parabola is a vertical line passing through it
∴ The equation of the axis of symmetry is x = h
- The vertex of the parabola could be minimum point if the parabola
opened upward or maximum if it opened downward
- The minimum value and the maximum value are the value of k
# Look to the attached figures for more understand
* Now lets solve the problem
∵ f(x) = x(x - 4)
∵ Point (2 + d , y) is on the graph of f(x)
- Replace each x in f(x) by 2 + d
∴ f(2 + d) = (2 + d)(2 + d - 4) ⇒ add 2 and -4
∴ f(2 + d) = (2 + d)(-2 + d)
∵ f(2 + d) = y
∴ y = (2 + d)( -2 + d)
* Multiply them to simplify
∴ y = 2(-2) + 2(d) + d(-2) + d(d) = -4 + 2d - 2d + d²
∴ y = -4 + d²
* Lets do these steps again with point (2 - d , y)
- Replace each x in f(x) by 2 - d
∴ f(2 - d) = (2 - d)(2 - d - 4) ⇒ add 2 and -4
∴ f(2 - d) = (2 - d)(-2 - d)
∵ f(2 - d) = y
∴ y = (2 - d)( -2 - d)
* Multiply them to simplify
∴ y = 2(-2) + 2(-d) - d(-2) - d(-d) = -4 - 2d + 2d + d²
∴ y = -4 + d²
- The value of y of the point (2 - d , y) = the value of y of the point on
the graph
∵ f(2 + d) = f(2 - d)
∵ The point (2 + d , y) is on the graph of f(x)
∴ The point (2 - d , y) is on the graph of f(x)
* It is true the point (2 - d, y) is also on the graph.
* To find the relation between the line x = 2 and the graph of f(x)
lets find the vertex of the parabola
- If f(x) = ax² + bx + c in the general form, where a, b , c are constant
then h = -b/2a, where h is the x-coordinate of the vertex point, a is
the coefficient of x² and b is the coefficient of x
∵ f(x) = x(x - 4) ⇒ multiply the bracket by x to put it in the general form
∴ f(x) = x² - 4x
- Find the value of a and b to find h
∵ a = 1 and b = -4
∵ h = -b/2a
∴ h = -(-4)/2(1) = 4/2 = 2
∴ The x-coordinate of the vertex point = 2
∵ The axis of symmetry of the parabola passing through the
vertex point
∴ The equation of the axis of symmetry of the parabola is x = 2
* The line x = 2 is the axis of symmetry of the graph of f(x)